{"id":1018,"date":"2023-06-22T01:45:05","date_gmt":"2023-06-22T01:45:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probability-distributions-learn-it-2\/"},"modified":"2025-05-16T02:15:27","modified_gmt":"2025-05-16T02:15:27","slug":"probability-distributions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probability-distributions-learn-it-2\/","title":{"raw":"Probability Distributions: Learn It 2","rendered":"Probability Distributions: Learn It 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Understand the concept of a probability distribution and its role in describing the behavior of a random variable.<\/li>\r\n\t<li>Describe the characteristics of probability distributions.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox example\"><strong><strong><strong>Probability Distribution for Boreal Owl Eggs<\/strong><\/strong><\/strong>\r\n<p><img class=\"aligncenter wp-image-5741 size-thumbnail\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22014504\/m6_probability_topic_6_1_m6_intro_probability_distributions_image1.png\" alt=\"Boreal owl\" width=\"150\" height=\"150\" \/><br \/>\r\nBoreal owls are common in Canada and Alaska. They are fairly small, averaging [latex]10[\/latex] inches in length and weighing from [latex]4[\/latex] to [latex]6[\/latex] oz. They often make their nests in woodpecker holes. The number of eggs in a boreal owl nest generally ranges from [latex]4[\/latex] to [latex]6[\/latex] eggs. Using relative frequencies from large field observations, we can estimate the probability of a nest containing a certain number of eggs.<\/p>\r\n<p>The variable is the <em>number of Boreal owl eggs in a nest<\/em>.<br \/>\r\nThis is a quantitative variable with values [latex]0, 1, 2, 3, 4, 5,[\/latex] or [latex]6[\/latex] eggs.<br \/>\r\nThe probability distribution gives the probability that a nest will have from [latex]0[\/latex] to [latex]6[\/latex] eggs.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>Number of Eggs<\/th>\r\n<th>Probability<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]0.2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]0.1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]0.1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]0.25[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]0.25[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]0.05[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]6[\/latex]<\/td>\r\n<td>[latex]0.05[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<br \/>\r\n<p><img class=\"aligncenter wp-image-3070 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/11201952\/newplot-6.png\" alt=\"A bar graph of the percentage of boreal owl eggs in a nest. The percentages match the probabilities in the above table.\" width=\"626\" height=\"350\" \/><\/p>\r\n<p>This table and bar graph are an example of a probability distribution. Each variable value is assigned a probability.<\/p>\r\n<p>Note:\u00a0The sum of all of the probabilities is [latex]1[\/latex]. This is always true for a probability distribution.<\/p>\r\n<p>We can use the probability distribution to answer probability questions:<\/p>\r\n<p>Question<strong>:<\/strong>\u00a0Which is more likely: (1) To find a boreal owl nest with [latex]3[\/latex] eggs, or (2) To find a boreal owl nest with [latex]4[\/latex] eggs.<\/p>\r\n<p>[reveal-answer q=\"513597\"]Show answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"513597\"]Answer: Both of these events are equally likely. [latex]P(3 eggs) = P(4 eggs) = 0.25[\/latex]. There is a [latex]25[\/latex]% chance that if you find a boreal owl nest, it will have [latex]3[\/latex] eggs. You are equally likely to find a boreal owl nest with [latex]4[\/latex] eggs.[\/hidden-answer]<\/p>\r\n<p>Question<strong>:<\/strong> Do the data points in the probability distribution follow a specific pattern or distribution shape, such as uniform, symmetric, or skewed? What can you say about the mean and median values of the probability distribution? How does this balance between the mean and median affect our understanding of the distribution's central tendencies and symmetry?<\/p>\r\n<p>[reveal-answer q=\"705771\"]Show answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"705771\"]The probability distribution is not uniform or symmetric. It is skewed left.<\/p>\r\n<p>In a left-skewed probability distribution, we find that the mean is less than the median. The mean is sensitive to outliers and tends to be pulled in the direction of the skew, while the median is resistant to extreme values.<\/p>\r\n<p>The relationship between the mean and median in a left-skewed distribution informs us about the distribution's central tendencies, emphasizing the impact of extreme values on the mean and the median's resistance to such outliers. It also highlights the distribution's lack of symmetry, which is essential for understanding the data's shape and characteristics.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section>The variable \"Number of eggs\" above is classified as a <strong>discrete<\/strong>\u00a0<strong>variable<\/strong> because it takes a fixed set of possible numerical values, and it is not possible to get any value in between.\u00a0\r\n\r\n<section class=\"textbox proTip\">A <strong>discrete probability distribution<\/strong> is a type of probability distribution that shows all possible values of a discrete random variable (countable or finite outcomes) along with the probabilities associated with those outcomes.<\/section>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand the concept of a probability distribution and its role in describing the behavior of a random variable.<\/li>\n<li>Describe the characteristics of probability distributions.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\"><strong><strong><strong>Probability Distribution for Boreal Owl Eggs<\/strong><\/strong><\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5741 size-thumbnail\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22014504\/m6_probability_topic_6_1_m6_intro_probability_distributions_image1.png\" alt=\"Boreal owl\" width=\"150\" height=\"150\" \/><br \/>\nBoreal owls are common in Canada and Alaska. They are fairly small, averaging [latex]10[\/latex] inches in length and weighing from [latex]4[\/latex] to [latex]6[\/latex] oz. They often make their nests in woodpecker holes. The number of eggs in a boreal owl nest generally ranges from [latex]4[\/latex] to [latex]6[\/latex] eggs. Using relative frequencies from large field observations, we can estimate the probability of a nest containing a certain number of eggs.<\/p>\n<p>The variable is the <em>number of Boreal owl eggs in a nest<\/em>.<br \/>\nThis is a quantitative variable with values [latex]0, 1, 2, 3, 4, 5,[\/latex] or [latex]6[\/latex] eggs.<br \/>\nThe probability distribution gives the probability that a nest will have from [latex]0[\/latex] to [latex]6[\/latex] eggs.<\/p>\n<table>\n<tbody>\n<tr>\n<th>Number of Eggs<\/th>\n<th>Probability<\/th>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0.2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]0.1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]0.1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]0.25[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]0.25[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]0.05[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]6[\/latex]<\/td>\n<td>[latex]0.05[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"wp-nocaption \"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3070 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/11201952\/newplot-6.png\" alt=\"A bar graph of the percentage of boreal owl eggs in a nest. The percentages match the probabilities in the above table.\" width=\"626\" height=\"350\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/11201952\/newplot-6.png 626w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/11201952\/newplot-6-300x168.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/11201952\/newplot-6-65x36.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/11201952\/newplot-6-225x126.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/11201952\/newplot-6-350x196.png 350w\" sizes=\"(max-width: 626px) 100vw, 626px\" \/><\/p>\n<p>This table and bar graph are an example of a probability distribution. Each variable value is assigned a probability.<\/p>\n<p>Note:\u00a0The sum of all of the probabilities is [latex]1[\/latex]. This is always true for a probability distribution.<\/p>\n<p>We can use the probability distribution to answer probability questions:<\/p>\n<p>Question<strong>:<\/strong>\u00a0Which is more likely: (1) To find a boreal owl nest with [latex]3[\/latex] eggs, or (2) To find a boreal owl nest with [latex]4[\/latex] eggs.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q513597\">Show answer<\/button><\/p>\n<div id=\"q513597\" class=\"hidden-answer\" style=\"display: none\">Answer: Both of these events are equally likely. [latex]P(3 eggs) = P(4 eggs) = 0.25[\/latex]. There is a [latex]25[\/latex]% chance that if you find a boreal owl nest, it will have [latex]3[\/latex] eggs. You are equally likely to find a boreal owl nest with [latex]4[\/latex] eggs.<\/div>\n<\/div>\n<p>Question<strong>:<\/strong> Do the data points in the probability distribution follow a specific pattern or distribution shape, such as uniform, symmetric, or skewed? What can you say about the mean and median values of the probability distribution? How does this balance between the mean and median affect our understanding of the distribution&#8217;s central tendencies and symmetry?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q705771\">Show answer<\/button><\/p>\n<div id=\"q705771\" class=\"hidden-answer\" style=\"display: none\">The probability distribution is not uniform or symmetric. It is skewed left.<\/p>\n<p>In a left-skewed probability distribution, we find that the mean is less than the median. The mean is sensitive to outliers and tends to be pulled in the direction of the skew, while the median is resistant to extreme values.<\/p>\n<p>The relationship between the mean and median in a left-skewed distribution informs us about the distribution&#8217;s central tendencies, emphasizing the impact of extreme values on the mean and the median&#8217;s resistance to such outliers. It also highlights the distribution&#8217;s lack of symmetry, which is essential for understanding the data&#8217;s shape and characteristics.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section>The variable &#8220;Number of eggs&#8221; above is classified as a <strong>discrete<\/strong>\u00a0<strong>variable<\/strong> because it takes a fixed set of possible numerical values, and it is not possible to get any value in between.\u00a0<\/p>\n<section class=\"textbox proTip\">A <strong>discrete probability distribution<\/strong> is a type of probability distribution that shows all possible values of a discrete random variable (countable or finite outcomes) along with the probabilities associated with those outcomes.<\/section>\n<\/section>\n","protected":false},"author":8,"menu_order":6,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":3053,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1018"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":19,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1018\/revisions"}],"predecessor-version":[{"id":6681,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1018\/revisions\/6681"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/3053"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1018\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1018"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1018"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1018"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1018"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}