Data shows that 88% of the people in a certain population are right-handed. A group of 7 people from this population are selected at random.
(a) What would be the expected value for the number of right-handed people in the group?
(b) What is the probability that exactly 5 of the people in the group are right-handed?
(c) What is the probability that at least 5 of the people in the group are right-handed?
(a) Take sample size times .88 for expected value.
(b) Use a binomial probability table. Values are: n = 7, x = 5, and p = .88
(c) Use the table again. Find x = 5,6,7. Use same values for n and p. Add together for total probability.
I hope this will help get you started.
To answer these questions, we can use the concept of probability and expected value. Let's break down each question and explain how to get the answers.
(a) Expected value for the number of right-handed people in the group:
The expected value is calculated by multiplying the probability of each outcome by the corresponding value, and then summing up these products. In this case, we know that 88% of the population is right-handed, so the probability of any individual being right-handed is 0.88. Since we are selecting 7 people at random, the expected value for the number of right-handed people in the group would be given by:
Expected value = Number of people selected x Probability of being right-handed = 7 x 0.88 = 6.16
(b) Probability that exactly 5 of the people in the group are right-handed:
To calculate this probability, we need to use the concept of binomial probability. Binomial probability is used when we have a fixed number of independent trials (in this case, selecting 7 people) and each trial only has two possible outcomes (right-handed or not right-handed). The probability of exactly 5 right-handed people can be calculated using the formula:
Probability = (Number of ways to choose 5 right-handed people out of 7) x (Probability of right-handed person)^5 x (Probability of not right-handed person)^(7-5)
To calculate the number of ways to choose 5 right-handed people out of 7, we use the combination formula: C(7, 5) = 7! / (5! * (7-5)!). Plugging the values into the formula, we get:
Probability = (C(7, 5)) * (0.88^5) * (0.12^2)
(c) Probability that at least 5 of the people in the group are right-handed:
To calculate this probability, we need to find the sum of the probabilities of exactly 5, 6, and 7 right-handed people. We can calculate each probability using the formula mentioned above for exactly 5 right-handed people, exactly 6 right-handed people, and exactly 7 right-handed people, and then add them together.
That's how you can calculate the expected value and probabilities for the given scenario.