A survey of motorcycle riders found that 21% of riders always wear a helmet. If 5 riders are chosen at random:(*Round answers to nearest ten thousandths of a percent)What is the probability that none of the chosen riders wears their helmet all of the time? %What is the probability that at least one of them does always wear a helmet?
Question
A survey of motorcycle riders found that 21% of riders always wear a helmet. If 5 riders are chosen at random:(*Round answers to nearest ten thousandths of a percent)What is the probability that none of the chosen riders wears their helmet all of the time? %What is the probability that at least one of them does always wear a helmet?
Solution
To solve this problem, we need to understand that it is a binomial probability problem.
- The probability that none of the chosen riders wears their helmet all of the time:
In this case, we want to find the probability that none of the 5 riders (n=5) always wears their helmet (k=0). The probability that a rider always wears their helmet is 21% or 0.21 (p=0.21). The probability that a rider does not always wear their helmet is therefore 1 - 0.21 = 0.79 (q=0.79).
The formula for binomial probability is:
P(X=k) = C(n, k) * (p^k) * (q^(n-k))
Where:
- P(X=k) is the probability we want to find
- C(n, k) is the combination of n items taken k at a time
- p is the probability of success
- q is the probability of failure
- n is the number of trials
- k is the number of successes
Plugging the values into the formula, we get:
P(X=0) = C(5, 0) * (0.21^0) * (0.79^5) = 1 * 1 * 0.41 = 0.41 or 41%
- The probability that at least one of them does always wear a helmet:
This is the complement of the probability that none of them always wears a helmet.
So, P(X>=1) = 1 - P(X=0) = 1 - 0.41 = 0.59 or 59%
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