Use a binomial distribution to calculate probability
Determine if a probability model meets the conditions for a binomial distribution
Bernoulli Trial
Flipping a coin is a classic example of a Bernoulli trial. When we flip a coin, there are two possible outcomes, heads or tails, and the different flips of the coin are independent. We can think of flipping tails as a “success,” and if the coin is fair, the probability of success is [latex]p = 0.5[/latex] for every trial.Let’s experiment flipping a coin [latex]3[/latex] times and counting the number of tails obtained.Flipping a coin [latex]3[/latex] times is a binomial experiment where [latex]n = 3 \text{ and } p = 0.5[/latex], and the random variable is the number of tails in [latex]3[/latex] coin flips. Therefore, the distribution of [latex]X[/latex] can be modeled using the binomial distribution.
Recall that the outcomes of the experiment are as given in the following table:
Experimental
Outcome
[latex]X[/latex]
Number of Tails in 3 Flips of a Coin
HHH
[latex]0[/latex]
HHT
[latex]1[/latex]
HTH
[latex]1[/latex]
THH
[latex]1[/latex]
TTH
[latex]2[/latex]
THT
[latex]2[/latex]
HTT
[latex]2[/latex]
TTT
[latex]3[/latex]
The probability of each value of [latex]X[/latex] can be found by calculating its relative frequencies.
Notice that since the trials here are independent, you can also find these probabilities by using the rule for independent events, [latex]P(A \mbox{ and } B) = P(A) \cdot P(B)[/latex], in combination with the rule for finding OR probabilities.
Notice that the probabilities you found in part (a) and (b) are the same. This is due to the fact that the trials are independent!