A coin is weighted so that the probability of obtaining a head in a single toss is 0.37. If the coin is tossed 27 times, find the following probabilities. Round your standard normal variable to two decimal places before using the table of values. (Give your answers to four decimal places.)
(a) more than 12 heads
(b) between 12 and 15 heads, inclusive
(c) less than 10 heads
To find the probabilities, we will use the binomial distribution formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
where:
- P(X = k) is the probability of getting exactly k successes (in this case, heads)
- C(n, k) is the number of combinations of n items taken k at a time
- p is the probability of success (getting heads)
- (1-p) is the probability of failure (getting tails)
- n is the number of trials (tosses)
Let's calculate each probability step by step:
(a) P(more than 12 heads):
We need to calculate the probability of getting 13, 14, 15, ..., 27 heads (inclusive) and then sum them up:
P(X > 12) = P(X = 13) + P(X = 14) + ... + P(X = 27)
Using the binomial distribution formula, we can calculate each term:
P(X = k) = C(27, k) * p^k * (1-p)^(27-k)
Substituting p = 0.37 and k = 13, 14, ..., 27, we can calculate each probability term. Then we sum them up to find P(X > 12).
(b) P(between 12 and 15 heads, inclusive):
We need to calculate the probability of getting 12, 13, 14, 15 heads and then sum them up:
P(12 ≤ X ≤ 15) = P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)
Using the binomial distribution formula, we can calculate each term:
P(X = k) = C(27, k) * p^k * (1-p)^(27-k)
Substituting p = 0.37 and k = 12, 13, 14, 15, we can calculate each probability term. Then we sum them up to find P(12 ≤ X ≤ 15).
(c) P(less than 10 heads):
We need to calculate the probability of getting 0, 1, 2, ..., 9 heads and then sum them up:
P(X < 10) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 9)
Using the binomial distribution formula, we can calculate each term:
P(X = k) = C(27, k) * p^k * (1-p)^(27-k)
Substituting p = 0.37 and k = 0, 1, 2, ..., 9, we can calculate each probability term. Then we sum them up to find P(X < 10).
By following these steps, you can calculate each probability using the binomial distribution formula.