63% of men consider themselves baseball fans. You randomly select 10 and ask if he consider him self a baseball fan. Find the probability that the number who consider themselves baseball fans is (a) exactly 8, (b) at least (8) less than (8)
Assuming the random selection is from a large pool of people, in which case the probability remains constant throughout the experiment.
This experiment could then be modeled by the binomial distribution, with probability
P(X=n)=C(N,x)px(1-p)N-x
where p=probability of success = 0.63
N=10 (sample size)
For part (a)
calculate P(X=8)
for part (b),
calculate ΣP(X=i) where i=8,9,10
for part (c),
calculate (1-answer in part (b))
since
ΣP(X=0.....10) = 1
To find the probability in this question, we can apply the binomial probability formula. The binomial probability formula calculates the probability of exactly 'x' successes in 'n' independent Bernoulli trials, where each trial has a fixed probability 'p' of success.
The formula is:
P(X = x) = (nCx) * p^x * (1 - p)^(n-x)
Where:
- P(X = x) is the probability of getting exactly x successes
- nCx is the number of combinations of choosing x successes in n trials (n choose x)
- p is the probability of success in one trial
- (1 - p) is the probability of failure in one trial
- x is the number of successes you are interested in
- n is the total number of trials
Now, let's calculate the probabilities for the given scenarios:
(a) Exactly 8 men consider themselves baseball fans:
n = 10 (total number of trials)
x = 8 (number of successes)
p = 0.63 (probability of success in one trial, as stated in the question)
Using the formula:
P(X = 8) = (10C8) * 0.63^8 * (1 - 0.63)^(10-8)
Calculating the values:
(10C8) = (10! / (8! * (10-8)!)) = 45
0.63^8 ≈ 0.046
(1 - 0.63)^(10-8) ≈ 0.169
P(X = 8) ≈ 45 * 0.046 * 0.169 ≈ 0.138 or 13.8%
Therefore, the probability that exactly 8 men consider themselves baseball fans is approximately 0.138 or 13.8%.
(b) At least 8 men consider themselves baseball fans:
To find the probability of "at least" 8 successes, we need to calculate the probabilities of 8, 9, and 10 successes separately and then sum them up.
P(X ≥ 8) = P(X = 8) + P(X = 9) + P(X = 10)
We already calculated P(X = 8) in part (a). Now let's calculate the other probabilities:
P(X = 9) = (10C9) * 0.63^9 * (1 - 0.63)^(10-9)
(10C9) = (10! / (9! * (10-9)!)) = 10
0.63^9 ≈ 0.029
(1 - 0.63)^(10-9) ≈ 0.63
P(X = 10) = (10C10) * 0.63^10 * (1 - 0.63)^(10-10)
(10C10) = (10! / (10! * (10-10)!)) = 1
0.63^10 ≈ 0.018
(1 - 0.63)^(10-10) = 1
Adding up the probabilities:
P(X ≥ 8) = 0.138 + (10 * 0.029 * 0.63) + (1 * 0.018 * 1)
P(X ≥ 8) ≈ 0.138 + 0.181 + 0.018 ≈ 0.337 or 33.7%
Therefore, the probability that at least 8 men consider themselves baseball fans is approximately 0.337 or 33.7%.