{"id":260,"date":"2024-10-18T21:15:25","date_gmt":"2024-10-18T21:15:25","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/applications-with-probability-learn-it-5\/"},"modified":"2024-10-18T21:15:25","modified_gmt":"2024-10-18T21:15:25","slug":"applications-with-probability-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebrademo\/chapter\/applications-with-probability-learn-it-5\/","title":{"raw":"Applications With Probability: Learn It 5","rendered":"Applications With Probability: Learn It 5"},"content":{"raw":"\n

Expected Value<\/h2>\n

Expected value<\/strong> is perhaps the most useful probability concept we will discuss. It has many applications, from insurance policies to making financial decisions, and it\u2019s one thing that the casinos and government agencies that run gambling operations and lotteries hope most people never learn about.<\/p>\n

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expected value<\/h3>\n

Expected Value<\/strong> is the average gain or loss of an event if the procedure is repeated many times.<\/p>\n

 <\/p>\n

We can compute the expected value by multiplying each outcome by the probability of that outcome, then adding up the products.<\/p>\n

 <\/p>\n

Notation:<\/strong> [latex]E(x)=\\sum_{i}^{n}P(x_i)x_i[\/latex]<\/p>\n<\/div>\n<\/section>\n

In a certain state's lottery, [latex]48[\/latex] balls numbered [latex]1[\/latex] through [latex]48[\/latex] are placed in a machine and six of them are drawn at random. If the six numbers drawn match the numbers that a player had chosen, the player wins [latex]$1,000,000[\/latex]. If they match [latex]5[\/latex] numbers, then win [latex]$1,000[\/latex].   It costs [latex]$1[\/latex] to buy a ticket. Find the expected value.
\n[reveal-answer q=\"737029\"]Show Solution[\/reveal-answer]
\n[hidden-answer a=\"737029\"]Earlier, we calculated the probability of matching all [latex]6[\/latex] numbers and the probability of matching [latex]5[\/latex] numbers:[latex]\\frac{{}_{6}{{C}_{6}}}{{}_{48}{{C}_{6}}}=\\frac{1}{12271512}\\approx0.0000000815[\/latex] for all [latex]6[\/latex] numbers,[latex]\\frac{\\left({}_{6}{{C}_{5}}\\right)\\left({}_{42}{{C}_{1}}\\right)}{{}_{48}{{C}_{6}}}=\\frac{252}{12271512}\\approx0.0000205[\/latex] for [latex]5[\/latex] numbers. Our probabilities and outcome values are:\n\n\n\n\n\n\n\n\n
Outcome<\/td>\nProbability of outcome<\/td>\n<\/tr>\n
[latex]$999,999[\/latex]<\/td>\n[latex]\\frac{1}{12271512}[\/latex]<\/td>\n<\/tr>\n
[latex]$999[\/latex]<\/td>\n[latex]\\frac{252}{12271512}[\/latex]<\/td>\n<\/tr>\n
[latex]-$1[\/latex]<\/td>\n[latex]1-\\frac{253}{12271512}=\\frac{12271259}{12271512}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The expected value, then is:<\/p>\n

[latex]\\left(\\$999,999 \\right)\\cdot \\frac{1}{12271512}+\\left( \\$999\\right)\\cdot\\frac{252}{12271512}+\\left(-\\$1\\right)\\cdot\\frac{12271259}{12271512}\\approx-\\$0.898[\/latex]<\/p>\n

On average, one can expect to lose about [latex]90[\/latex] cents on a lottery ticket. Of course, most players will lose [latex]$1[\/latex].<\/p>\n

View more about the expected value examples in the following video.<\/p>\n

[embed]https:\/\/youtu.be\/pFzgxGVltS8[\/embed]<\/p>\n

You can view the transcript for \u201cExpected value\u201d here (opens in new window).<\/a><\/p>\n

[\/hidden-answer]<\/p>\n<\/section>\n

[ohm2_question hide_question_numbers=1]3288[\/ohm2_question]<\/section>\n

The sign of the expected value can tell different things about a situation. Let's take gambling as an example. In general, if the expected value of a game is negative, it is not a good idea to play the game, since on average you will lose money. It would be better to play a game with a positive expected value (good luck trying to find one!), although keep in mind that even if the average<\/em> winnings are positive it could be the case that most people lose money and one very fortunate individual wins a great deal of money.  If the expected value of a game is [latex]0[\/latex], we call it a fair game<\/em>, since neither side has an advantage.<\/p>\n

Expected value also has applications outside of gambling, it is very common in making insurance decisions. When it comes to insurance, insurance companies want the expected value to be negative as well. The insurance company can only afford to offer policies if they, on average, make money on each policy. They can afford to pay out the occasional benefit because they offer enough policies that those benefit payouts are balanced by the rest of the insured people. Let's try an insurance example.<\/p>\n

A [latex]40[\/latex]-year-old man in the U.S. has a [latex]0.242\\%[\/latex] risk of dying during the next year.[footnote]According to the estimator at http:\/\/www.numericalexample.com\/index.php?view=article&id=91[\/footnote] An insurance company charges [latex]$275[\/latex] for a life-insurance policy that pays a [latex]$100\\mbox{,}000[\/latex] death benefit. What is the expected value for the person buying the insurance?
\n[reveal-answer q=\"90556\"]Show Solution[\/reveal-answer]
\n[hidden-answer a=\"90556\"]The probabilities and outcomes are\n\n\n\n\n\n\n\n
Outcome<\/td>\nProbability of outcome<\/td>\n<\/tr>\n
[latex]$100,000 - $275 = $99,725[\/latex]<\/td>\n[latex]0.00242[\/latex]<\/td>\n<\/tr>\n
[latex]-$275[\/latex]<\/td>\n[latex]1 \u2013 0.00242 = 0.99758[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

<\/p>\n

<\/p>\n

The expected value is [latex]($99\\mbox{,}725)(0.00242) + (-$275)(0.99758) = -$33[\/latex].<\/p>\n

[\/hidden-answer]<\/p>\n<\/section>\n","rendered":"

Expected Value<\/h2>\n

Expected value<\/strong> is perhaps the most useful probability concept we will discuss. It has many applications, from insurance policies to making financial decisions, and it\u2019s one thing that the casinos and government agencies that run gambling operations and lotteries hope most people never learn about.<\/p>\n

\n
\n

expected value<\/h3>\n

Expected Value<\/strong> is the average gain or loss of an event if the procedure is repeated many times.<\/p>\n

 <\/p>\n

We can compute the expected value by multiplying each outcome by the probability of that outcome, then adding up the products.<\/p>\n

 <\/p>\n

Notation:<\/strong> [latex]E(x)=\\sum_{i}^{n}P(x_i)x_i[\/latex]<\/p>\n<\/div>\n<\/section>\n

In a certain state’s lottery, [latex]48[\/latex] balls numbered [latex]1[\/latex] through [latex]48[\/latex] are placed in a machine and six of them are drawn at random. If the six numbers drawn match the numbers that a player had chosen, the player wins [latex]$1,000,000[\/latex]. If they match [latex]5[\/latex] numbers, then win [latex]$1,000[\/latex].   It costs [latex]$1[\/latex] to buy a ticket. Find the expected value.<\/p>\n