A tennis player makes a successful first serve 51% of the time. if she serves 9 times, what is the probability that she gets exactly 3 first serves in? assume that each serve is independent of the others.

0.154229089

N=9 p=.51 q=1-.51=.49

9 nCr 3= 84
84(.51^3)(.49^6)
=0.154

Well, it seems this tennis player has a 51% chance of making a successful first serve. If we assume that each serve is independent of the others, we can use the binomial probability formula to calculate the probability of getting exactly 3 first serves in.

The formula is: P(x) = (nCx) * (p^x) * ((1-p)^(n-x))

Where:
P(x) is the probability of getting exactly x successful first serves,
n is the total number of serves (9 in this case),
x is the number of successful first serves (3 in this case),
p is the probability of making a successful first serve (51% or 0.51 in this case),
(1-p) is the probability of not making a successful first serve (49% or 0.49 in this case),
and (nCx) represents the binomial coefficient, which can be calculated as n! / (x!(n-x)!).

So, plugging in the values, we have:
P(3) = (9C3) * (0.51^3) * (0.49^6)

Now, let me calculate that for you. *Calculating...*

Ta-da! The probability of the tennis player getting exactly 3 first serves in is approximately 0.278 or 27.8%. But hey, even if the odds aren't always in her favor, she can always serve up some laughs on the court!

To find the probability of getting exactly 3 first serves out of 9, we can use the binomial probability formula.

The binomial probability formula is given by:

P(X = k) = (nCk) * p^k * (1 - p)^(n - k)

Where:
n = number of trials (in this case, 9 serves)
k = number of successful outcomes (in this case, 3 first serves)
p = probability of success on a single trial (in this case, 51%, or 0.51)
(1 - p) = probability of failure on a single trial (in this case, 49%, or 0.49)

Now, let's plug the values into the formula:

P(X = 3) = (9C3) * (0.51)^3 * (0.49)^(9 - 3)

The value of (nCk) represents the number of combinations of n items taken k at a time, and can be calculated using the formula:

(nCk) = n! / (k! * (n - k)!)

Let's calculate the individual terms and substitute them into the formula:

(9C3) = 9! / (3! * (9 - 3)!) = (9 * 8 * 7) / (3 * 2 * 1) = 84

(0.51)^3 = 0.132651

(0.49)^(9 - 3) = 0.005336096

Plugging these values into the formula, we get:

P(X = 3) = 84 * 0.132651 * 0.005336096

Calculating this expression, we find:

P(X = 3) ≈ 0.13805

Therefore, the probability that the tennis player gets exactly 3 first serves in out of 9 is approximately 0.13805, or 13.81%.

probability is constant and independent of trials.

trials are Bernouli, i.e. either success or failure.
Number of trials is known.
All this point to the binomial distribution where
P(k successes out of n)
=nCk p^k (1-p)^(n-k)
p=probability of success of each independent trial=0.51
n=total number of trials=9
k=3