If the probability that a certain tennis player will serve an ace is 1/4, what is the probability that he will serve exactly two aces out of four serves? (Assume that the four serves are independent. Round your answer to four decimal places.)
I guess I will have to use binomial coefficients.
the probability of k successes in n trials is:
P(k) = C(n,k) p^k (1-p)^(n-k)
C(n,k) is binomial coef
get from Pascal triangle or table or calculate from
C(n,k) - n! / [ k! (n-k)! ]
here
p = prob of ace = .25
(1-p) = .75
n = 4
k = 2
C(4,2) = 4! /[2!(2!)] = 4*3*2/[2(2)] = 6
so
P(2) = 6 (.25)^2 (.75)^2
= .21
To find the probability that the player will serve exactly two aces out of four serves, we can use the binomial probability formula.
The binomial probability formula is:
P(X=k) = (n choose k) * p^k * (1-p)^(n-k)
where:
- P(X=k) is the probability of getting exactly k successes
- n is the total number of trials (in this case, number of serves)
- k is the number of successes (in this case, number of aces)
- (n choose k) is the number of combinations of n items taken k at a time
- p is the probability of success on a single trial (in this case, probability of serving an ace)
- (1-p) is the probability of failure on a single trial
In this case, n = 4 (number of serves), k = 2 (number of aces), and p = 1/4 (probability of serving an ace).
Let's calculate the probability:
P(X=2) = (4 choose 2) * (1/4)^2 * (1-1/4)^(4-2)
Using the combinations formula:
(4 choose 2) = 4! / (2! * (4-2)!) = 6
Plugging in the values:
P(X=2) = 6 * (1/4)^2 * (3/4)^2
Calculating:
P(X=2) = 6 * (1/16) * (9/16) = 54/256
Simplifying:
P(X=2) = 0.2109 (rounded to four decimal places)
Therefore, the probability that the player will serve exactly two aces out of four serves is approximately 0.2109.
To find the probability that the tennis player will serve exactly two aces out of four serves, we can use the binomial probability formula.
The binomial probability formula is:
P(X = k) = (nCk) * (p^k) * ((1-p)^(n-k))
Where:
P(X = k) represents the probability of having exactly k successes,
nCk represents the number of ways to choose k successes out of n trials,
p represents the probability of success on a single trial,
n represents the total number of trials, and
k represents the number of desired successes.
In this case, the probability of serving an ace is 1/4, so p = 1/4. The total number of serves is 4, so n = 4. We want exactly two aces, so k = 2.
Substituting these values into the formula, we get:
P(X = 2) = (4C2) * ((1/4)^2) * ((1-(1/4))^(4-2))
Calculating each part separately:
4C2 = 4! / (2! * (4-2)!) = 6
(1/4)^2 = 1/16
(1-(1/4))^(4-2) = 9/16
Now, substitute the results back into the formula:
P(X = 2) = 6 * (1/16) * (9/16)
Calculate these values:
P(X = 2) = 54/256
Finally, round the answer to four decimal places:
P(X = 2) ≈ 0.2109
Therefore, the probability that the tennis player will serve exactly two aces out of four serves is approximately 0.2109.