Suppose the number of cell phones in a household has a binomial distribution
with parameters n = 17, and p = 40 %.
Find the probability of a household having (Keep at least 4 digits)
(a) 11 or 13 cell phones?
(b) 11 or fewer cell phones ?
(c) 14 or more cell phones ?
(d) fewer than 13 cell phones ?
(e) more than 11 cell phones ?
(a) 0.2545
(b) 0.7372
(c) 0.2628
(d) 0.4827
(e) 0.5173
To find the probabilities for each scenario, we can use the binomial probability formula:
P(X = k) = nCk * p^k * (1-p)^(n-k)
Where:
P(X = k) is the probability of getting exactly k successes.
n is the number of trials (in this case, the number of households).
k is the number of successes (the number of cell phones in a household).
p is the probability of success (the probability of having a cell phone in a household).
nCk is the binomial coefficient, also known as "n choose k" (the number of ways to choose k phones from n).
Now, let's calculate the probabilities for each scenario:
(a) Probability of a household having 11 or 13 cell phones:
P(X = 11) + P(X = 13) = 17C11 * (0.4)^11 * (1-0.4)^(17-11) + 17C13 * (0.4)^13 * (1-0.4)^(17-13)
(b) Probability of a household having 11 or fewer cell phones:
P(X ≤ 11) = P(X = 0) + P(X = 1) + ... + P(X = 11)
Calculate each individual probability and sum them up.
(c) Probability of a household having 14 or more cell phones:
P(X ≥ 14) = P(X = 14) + P(X = 15) + ... + P(X = 17)
Calculate each individual probability and sum them up.
(d) Probability of a household having fewer than 13 cell phones:
P(X < 13) = P(X = 0) + P(X = 1) + ... + P(X = 12)
Calculate each individual probability and sum them up.
(e) Probability of a household having more than 11 cell phones:
P(X > 11) = P(X = 12) + P(X = 13) + ... + P(X = 17)
Calculate each individual probability and sum them up.
By using the binomial probability formula, you can calculate the probabilities for each scenario. Note that the binomial coefficient (nCk) can be calculated as n! / (k!(n-k)!), where "!" represents the factorial function.
To find the probability of each scenario, we can use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
where P(X = k) is the probability of getting exactly k successes in n trials, n is the number of trials, p is the probability of success, and (n choose k) is the binomial coefficient.
(a) To find the probability of a household having 11 or 13 cell phones, we can calculate the individual probabilities and add them together:
P(X = 11) = (17 choose 11) * (0.40)^11 * (0.60)^(17-11)
P(X = 13) = (17 choose 13) * (0.40)^13 * (0.60)^(17-13)
Therefore, P(11 or 13 cell phones) = P(X = 11) + P(X = 13).
(b) To find the probability of a household having 11 or fewer cell phones, we sum up the probabilities of having each possible number of cell phones up to 11:
P(X ≤ 11) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 11)
(c) To find the probability of a household having 14 or more cell phones, we sum up the probabilities of having each possible number of cell phones from 14 up to the total number of trials (17 in this case):
P(X ≥ 14) = P(X = 14) + P(X = 15) + ... + P(X = 17)
(d) To find the probability of a household having fewer than 13 cell phones, we sum up the probabilities of having each number of cell phones from 0 to 12:
P(X < 13) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 12)
(e) To find the probability of a household having more than 11 cell phones, we sum up the probabilities of having each number of cell phones from 12 up to the total number of trials (17 in this case):
P(X > 11) = P(X = 12) + P(X = 13) + ... + P(X = 17)