Question

Every day Ahmet buys a scratch-off lottery ticket with a 40% chance of winning some prize. He noticed that whenever he wears his red shirt he usually wins. He decided to keep track of his winnings while wearing the shirt and found that he won 3 out of 3 times. Let's test the hypothesis that Ahmet's chance of winning while wearing the shirt is 40% as always versus the alternative that the chance is somehow greater. Assuming the hypothesis is correct, what is the probability of Ahmet winning 3 times out of 3? Round your answer, if necessary, to the nearest tenth of a percent.

a. 64.8%
b. 0.4%
c. 40.0%
d. 0.0%

Answer 1

The probability of Ahmet winning 3 times out of 3 assuming the hypothesis is correct is approximately 6.4%.

Step-by-step explanation:

To test the hypothesis that Ahmet's chance of winning while wearing the shirt is 40%, we need to find the probability of him winning 3 times out of 3 assuming the hypothesis is correct.

The probability of Ahmet winning a scratch-off lottery ticket with a 40% chance of winning some prize is calculated using the binomial probability formula. For each individual ticket, the probability of winning is 40% or 0.4. Since Ahmet won 3 times out of 3, we need to find the probability of getting exactly 3 successes in 3 trials.

Using the binomial probability formula, the probability of getting exactly 3 successes in 3 trials is given by:

P(X=3) = (nCk) * (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.

Substituting the values into the formula, we have:

P(X=3) = (3C3) * (0.4^3) * ((1-0.4)^(3-3))

P(X=3) = 1 * 0.064 * 1

P(X=3) = 0.064

Rounding to the nearest tenth of a percent, the probability of Ahmet winning 3 times out of 3 assuming the hypothesis is correct is approximately 6.4%.