Final answer:
To find the probability and expected number of 12-year-olds needed to find the first one who can pick out Colorado on a map, we can use the geometric distribution. The probability of sampling exactly five 12-year-olds is approximately 4.47%, the probability of sampling five or more is approximately 23.53%, and the expected number of students sampled is about 4.17.
Step-by-step explanation:
a. To find the probability of sampling exactly five 12-year-olds to find the first one who can pick out Colorado on a map, we can use the geometric distribution. The probability of success is 24% or 0.24. The probability of needing exactly five trials is given by the formula P(X = k) = (1 - p)^(k - 1) * p, where p is the probability of success and k is the number of trials.
Plugging in the values, we get P(X = 5) = (0.76)^4 * 0.24 ≈ 0.0447 or 4.47%.
b. To find the probability of sampling 5 or more 12-year-olds to find the first one who can pick out Colorado on a map, we need to find the complement of the probability of finding the first one in four or fewer trials. So, P(X ≥ 5) = 1 - P(X ≤ 4). Using the geometric distribution formula, we can calculate P(X ≤ 4) = 1 - (1 - p)^4
= 1 - 0.76^4
≈ 0.7647 or 76.47%. T
herefore, P(X ≥ 5) = 1 - 0.7647 ≈ 0.2353 or 23.53%.
c. To find the expected number of 12-year-olds you must sample until you find the first one who can pick out Colorado on a map, we can use the formula E(X) = 1 / p, where p is the probability of success. In this case, E(X) = 1 / 0.24 ≈ 4.17. Therefore, you can expect to sample about 4.17 12-year-olds until you find the first one who can pick out Colorado on a map.