Question

Suppose that 25% of all voters prefer Candidate A. If 10 people are chosen at random for a poll, what is the probability that fewer than 9 of them favor Candidate A? Probability (Please show your answer to 4 decimal places) Suppose that 20% of all voters prefer Candidate A. If 9 people are chosen at randorn for a poll, what is the probability that at least 7 of them favor Candidate A? Probability = (Please show your answer to 4 decimal places)

Answer 1

To find the probability that fewer than 9 out of 10 people favor Candidate A, we can use the binomial distribution formula. To find the probability that at least 7 out of 9 people favor Candidate A, we can also use the binomial distribution formula.

Step-by-step explanation:

To find the probability that fewer than 9 out of 10 people favor Candidate A, we can use the binomial distribution. The probability of each individual in the sample favoring Candidate A is 25%, which means the probability of not favoring Candidate A is 75%. So, the probability of fewer than 9 people favoring Candidate A can be calculated using the binomial formula:

P(X < 9) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 8)

Using this formula, we can calculate the probability:

P(X < 9) =
`(0.75^10) + (10 * 0.25 * 0.75^9) + (45 * 0.25^2 * 0.75^8) + (120 * 0.25^3 * 0.75^7) + (210 * 0.25^4 * 0.75^6) + (252 * 0.25^5 * 0.75^5) + (210 * 0.25^6 * 0.75^4) + (120 * 0.25^7 * 0.75^3) + (45 * 0.25^8 * 0.75^2)`

Calculating this expression gives us the probability that fewer than 9 people favor Candidate A.

Using the same approach, to find the probability that at least 7 out of 9 people favor Candidate A, we can use the binomial distribution again:

P(X >= 7) = P(X = 7) + P(X = 8) + P(X = 9)

Using the binomial formula, we can calculate the probability:

P(X >= 7) =
`(0.25^7 * 0.75^2)`+ (9 *
`0.25^8`* 0.75) + (10 *
`0.25^9`)

Calculating this expression gives us the probability that at least 7 people favor Candidate A.