1. 52% of adults say chocolate chip is their favorite cookie. 40 adults are randomly selected and ask their favorite cookie. Find the probability that
a) at most 15 people say chocolate chip is their favorite cookie.
b) at least 15 people say chocolate chip is their favorite cookie.
c) more than 15 people say chocolate chip is their favorite cookie.
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To find the probability for each scenario, we will use the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of k successes
- n is the number of trials
- p is the probability of success on a single trial
- C(n, k) is the number of combinations of n items taken k at a time
In this case, our trials are the 40 adults randomly selected, and the probability of success (choosing chocolate chip as their favorite cookie) is 52%.
a) For "at most 15 people say chocolate chip is their favorite cookie," we can use the cumulative binomial probability:
P(X ≤ 15) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 15)
To calculate this, we substitute the values into the formula for each value of k from 0 to 15, and then sum them up.
b) For "at least 15 people say chocolate chip is their favorite cookie," we can calculate the complement of the probability that fewer than 15 people say chocolate chip is their favorite cookie:
P(X ≥ 15) = 1 - P(X < 15)
To calculate this, we need to find the cumulative binomial probability of "less than 15," subtract that value from 1, and then use the complementary probability property.
c) For "more than 15 people say chocolate chip is their favorite cookie," we can calculate the complement of the probability that 15 or fewer people say it is their favorite cookie:
P(X > 15) = 1 - P(X ≤ 15)
To calculate this, we need to find the cumulative binomial probability of "less than or equal to 15," subtract that value from 1, and then use the complementary probability property.
By substituting the appropriate values into the binomial probability formula and performing the calculations, we can find the answers to each of the scenarios.