Answer:
a) .273
b) .468
c) .184
Explanation:
formula: binomial probability mass function
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
X is the random variable representing the number of successes.
k is the number of successes we're interested in
n is the total number of trials
p is the probability of success on each trial
(n choose k) represents the number of ways to choose k successes from n trials
n = 7, p = 0.35, and we want to find P(X = 3):
Binomial Probability Distribution Probability
a) To calculate the probability of exactly three successes, we use the formula for the binomial probability mass function:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
P(X = 3) = (7 choose 3) * 0.35^3 * (1 - 0.35)^(7 - 3)
= 0.273
(n choose k) * p^k * (1 - p)^(n - k)
(7 choose 3) * 0.35^3 * (1 - 0.35)^(7 - 3) = .273
Therefore, the probability of exactly three successes is 0.273.
b) To calculate the probability of less than three successes, we need to find the sum of the probabilities of zero, one, and two successes:
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
P(X = 0) = (7 choose 0) * 0.35^0 * (1 - 0.35)^(7 - 0) = 0.027
P(X = 1) = (7 choose 1) * 0.35^1 * (1 - 0.35)^(7 - 1) = 0.146
P(X = 2) = (7 choose 2) * 0.35^2 * (1 - 0.35)^(7 - 2) = 0.295
P(X < 3) = 0.027 + 0.146 + 0.295
= 0.468
Therefore, the probability of less than three successes is 0.468.
c) To calculate the probability of five or more successes, we need to find the sum of the probabilities of five, six, and seven successes:
P(X >= 5) = P(X = 5) + P(X = 6) + P(X = 7)
P(X = 5) = (7 choose 5) * 0.35^5 * (1 - 0.35)^(7 - 5) = 0.148
P(X = 6) = (7 choose 6) * 0.35^6 * (1 - 0.35)^(7 - 6) = 0.034
P(X = 7) = (7 choose 7) * 0.35^7 * (1 - 0.35)^(7 - 7) = 0.002
P(X >= 5) = 0.148 + 0.034 + 0.002
= 0.184
Therefore, the probability of five or more successes is 0.184
This kind of formula can help with elections
Elections: The binomial distribution can be used to calculate the
probability of a candidate winning a certain number of votes,
given the number of voters &
candidate's level of support.
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