A game at a fundraiser has a probability of 0.2 that she wins. She plans to play 5 times. If $p$ is the probability that she wins exactly 1 time, what is $p$? Enter your answer to the nearest hundredth.
Answer (1)
Identify the values: n = 1 , k = 1 , p = 0.2 .
Apply the binomial probability formula: P ( X = k ) = ( k n ) ∗ p k ∗ ( 1 − p ) ( n − k ) .
Calculate the binomial coefficient: ( 1 1 ) = 1 .
Calculate the probability: P ( X = 1 ) = 1 ∗ ( 0.2 ) 1 ∗ ( 0.8 ) 0 = 0.2 .
The probability of winning exactly 1 time is 0.2 .
Explanation
Understand the problem and provided data We are given a binomial probability problem where the probability of winning a game is 0.2. We want to find the probability of winning exactly 1 time. The formula for binomial probability is given as P ( X = k ) = ( k n ) ∗ p k ∗ ( 1 − p ) ( n − k ) , where n is the number of trials, k is the number of successes, and p is the probability of success.
Identify the values for n, k, and p In this case, the number of trials is n = 1 , the number of successes is k = 1 , and the probability of success is p = 0.2 .
Plug the values into the formula We plug the values of n, k, and p into the binomial probability formula: P ( X = 1 ) = ( 1 1 ) ∗ ( 0.2 ) 1 ∗ ( 1 − 0.2 ) ( 1 − 1 )
Calculate the binomial coefficient Calculate the binomial coefficient ( 1 1 ) = 1 ! ( 1 − 1 )! 1 ! = 1 ! 0 ! 1 ! = 1 ∗ 1 1 = 1
Calculate the probability P(X=1) Substitute the binomial coefficient and the values of p, k, and n into the formula: P ( X = 1 ) = 1 ∗ ( 0.2 ) 1 ∗ ( 0.8 ) 0 = 1 ∗ 0.2 ∗ 1 = 0.2
State the final answer Therefore, the probability of winning exactly 1 time is 0.2.
Examples
Consider a scenario where you are selling raffle tickets at a fundraiser. Each ticket has a 20% chance of winning a prize. If a person buys only one ticket, the probability that they win exactly one prize is 20%, or 0.2. This is a simple application of binomial probability where n=1 and k=1.
