Of 20 people invited to a pool party, 4 prefer vanilla ice cream, 7
prefer chocolate, and 3 prefer strawberry. The host surveys six of these
people at random to determine how much ice cream to buy.
a) What is the probability that at least 3 of the people surveyed prefer
chocolate ice cream?
b) What is the probability that none prefer vanilla ice cream?
c) What is the expected number of people who prefer strawberry ice
cream?
d) What is the expected number of people who do not have a
preference for any of the three flavours?
I got 0.353 but the answer says it 0.3359, why am I wrong?
You got 0.353 because of the decimals they were rounded up, so technically your answer is correct, if your teacher is decent than he would mark your answer right because its close enough.
To calculate the probabilities in this problem, we can use the concept of combinations.
a) To find the probability that at least 3 of the people surveyed prefer chocolate ice cream, we need to consider the different ways this can occur. Since there are 7 people who prefer chocolate, we can select 3, 4, 5, or 6 of them in the sample of 6 people.
To calculate the probability, we need to divide the favorable outcomes by the total possible outcomes. The favorable outcomes are the combinations of selecting 3, 4, 5, or 6 people who prefer chocolate from the total number of people who prefer chocolate (7). The total possible outcomes are the combinations of selecting any 6 people from the total number of people invited (20).
The probability can be calculated as follows:
P(at least 3 people prefer chocolate) = (C(3,7) + C(4,7) + C(5,7) + C(6,7)) / C(6,20)
Using a calculator, we can find:
P(at least 3 people prefer chocolate) = (35 + 35 + 21 + 7) / 38,760
P(at least 3 people prefer chocolate) = 98 / 38,760
P(at least 3 people prefer chocolate) ≈ 0.002527
Therefore, the probability that at least 3 of the people surveyed prefer chocolate ice cream is approximately 0.002527, not 0.353.
Now let's move on to part b), c), and d):
b) To find the probability that none of the people surveyed prefer vanilla ice cream, we need to select 6 people who prefer flavors other than vanilla from the total number of people who do not prefer vanilla (20 - 4 = 16).
P(none prefer vanilla ice cream) = C(6,16) / C(6,20)
Using a calculator, we can find:
P(none prefer vanilla ice cream) = 8008 / 38,760
P(none prefer vanilla ice cream) ≈ 0.20632
Therefore, the probability that none of the people surveyed prefer vanilla ice cream is approximately 0.20632.
c) To find the expected number of people who prefer strawberry ice cream, we can multiply the probability of an individual preferring strawberry ice cream (3 out of 20) by the number of people in the sample (6).
Expected number of people who prefer strawberry ice cream = 3/20 * 6
Expected number of people who prefer strawberry ice cream = 0.3
Therefore, the expected number of people who prefer strawberry ice cream is 0.3.
d) To find the expected number of people who do not have a preference for any of the three flavors, we subtract the expected number of people who prefer vanilla, chocolate, or strawberry from the total number of people in the sample (6).
Expected number of people who do not have a preference = 6 - (4/20 * 6 + 7/20 * 6 + 3/20 * 6) = 6 - (1.2 + 2.1 + 0.9) = 6 - 4.2 = 1.8
Therefore, the expected number of people who do not have a preference for any of the three flavors is 1.8.
It seems that your answer for part a) is incorrect, and the probability should be much smaller than 0.353. Check your calculations and ensure you calculate the combinations correctly.