Question

a certain tennis player makes a successful first serve 73% of the time. assume that each serve is independent of the others. if she serves 6 times, what's the probability she gets a)all 6 serves in? b) exactly 2 serves in? c) at least 4 serves in? d) no more than 4 serves in?

Answer 1

a) Use Bernouli's Theorem.
\[P(6)=\left(\begin{matrix}6 \\6\end{matrix}\right)(0.73)^{6}=(0.73)^{6}\]

Answer 2

b) Use Bernouli's Theorem again.
\[P(2)=\left(\begin{matrix}6 \\ 2\end{matrix}\right)(0.73)^{2}(1-0.73)^{6-2}= \frac{6\times 5}{2}(0.73)^{2}(0.27)^{4}=?\]

Can you calculate the answers to a) and b) ?

Answer 3

yes i can, so to get all 6 serves in, it would be equal to .1513342263

Answer 4

and to get exactly 2 serves in would be .0012483502

Answer 5

wait i made a mistake for the 2 serves one, one second

Answer 6

actually, i'm a bit confused. where did you get the \[(6x5)/2\] from?

Answer 7

\[\frac{6\times 5}{2}\]
comes from calculation of the binomial coefficient
\[\left(\begin{matrix}6 \\ 2\end{matrix}\right)=\frac{6!}{2!(6-2)!}=\frac{6!}{2!4!}=\frac{6\times 5}{2}\]

Answer 8

but \[6!\] is equal to 6 x 5 x 4 x 3 x 2 x 1, how come it's just 6 x 5 there?

Answer 9

\[\frac{6\times 5\times 4\times 3\times 2\times 1}{2\times 1\times 4\times 3\times 2\times 1}=\frac{6\times 5}{2}\]
Notice how the last 4 terms of factorial 6 cancel with the 4 terms of factorial 4.

Answer 10

ohhh, i see. thank you so much, you helped alot. :)

Answer 11

Good! Well done. Shall we continue?

Answer 12

yes, i'm calculating for the exactly 2 serves question now :)

Answer 13

so for 2 serves, the probability is .0424807263

Answer 14

so to find the probability that she gets at least 4 serves, would we need to find the probability of her getting 4 serves, 5 serves, and 6 serves? and add them all up

Answer 15

Good work :). You have the correct answers for a) and b). Now to tackle c).
To find the probability of getting at least 4 first serves in we need to find the probabilities of getting exactly 4 serves in, 5 serves in and six serves in and sum the 3 probabilities.
We already have P(6), so we need to find P(4) and P(5)
\[P(4)=\left(\begin{matrix}6 \\ 4\end{matrix}\right)(0.73)^{4}(0.27)^{2}=\frac{6\times 5}{2}(0.73)^{4}(0.27)^{2}\]
What do you get for P(4) ?

Answer 16

for P(4)= .3105347653 and for P(5)= .3358375981

Answer 17

and i add the 3 probabilities together and i get .7977065897

Answer 18

Excellent work! You are really grasping this very well. All your answers are all correct so far. What are your thoughts on d) ?

Answer 19

for d, i would find the probabilities for P(0) + P(1) + P(2) + P(3) + P(4)

Answer 20

That's right. You already have P(2) and P(4). Do you need equations for the others or can you try finding them yourself?

Answer 21

for P(0) though, when i'm doing the factorial part i get \[(6!)/0!(6-0)!\]

Answer 22

will i get an undefined answer since 0! is 0 making the denominator 0

Answer 23

Very good question. For the purposes of statistics 0! has been defined as 1.

Answer 24

ohh i see. thank you SO SO much, i've been struggling with this in class but i finally understand now. thank you! :)

Answer 25

That's marvellous. I'm so happy to have been able to help :)

Answer 26

oh, one last thing. is there a shortcut in simplifying the factorial part right away? for example, how you went from \[6!/2(6-2)!\] to (6x5)/2

Answer 27

because each time i did it, i wrote out all the number's and then cancelled it when i wrote everything out

Answer 28

Yes there is a shortcut.
6! = 6 * 5 * 4!
If you split up 6! as above you can see right away how dividing by 4! gives 6 * 5

Answer 29

oh yes. i see it now. wow, makes life a bit simpler haha, thank you.
also, sorry to be a bother. when i was calculating the probability for getting 0 serves in, the equation i used was (.73)^0 x (.27)^6

Answer 30

(.73)^0 = 1 but when i entered (.27)^6 in the calculator, i get 3.87420489 E-4

Answer 31

i don't quite understand what that E-4 is

Answer 32

Your calculator is set up to give the answer in scientific notation.
3.87420489 E-4 = 3.87420489 * 10^-4 = 0.000387420489

Answer 33

E stands for exponent. In this case the exponent of 10.

Answer 34

ohh, got it. i used a graphing calculator when i got that E-4 but when i used a regular scientific calculator, i got the .000387420489. wow, really, thank you so very much for helping me through all these parts for this problem. i really appreciate it :)

Answer 35

It's good to clear up all these things now :)