00:01
A tennis player makes a successful serve 77 % of the time.
00:05
And we're looking at four independent serves.
00:08
So n is 4.
00:09
And p probability that she makes a successful first serve is 0 .77%.
00:16
So because we have four independent trials, two outcomes, successful or not, same probability on each serve, we have a binomial distribution.
00:27
So we can use the binomial formula for this question, and i'll need to do that.
00:34
The binomial formula is the probability of exactly x successes, is n choose x, p to the x, 1 minus p to the n minus x.
00:45
We don't really need it for part a, because part a is the same thing happening four times in a row.
00:51
Because these are independent, we can combine their probabilities by multiplication.
00:56
So it's the probability of success multiplied by itself for the four attempts.
01:04
0 .77 to the power of 4, 0 .3515.
01:12
So you could use the formula, but the first term would just be a 1, the last term will just be a 1 as well.
01:19
Next for part b, exactly two serves.
01:25
So two successes, two failures.
01:29
This term is for the 2, 3.
01:31
Successes.
01:35
Not .77 multiplied by itself.
01:38
This term is for the two failures.
01:41
The rest of the time, the other 23 % of the time, again twice.
01:47
This is the probability.
01:48
The first two serves are successful, the last two are not.
01:52
That's two out of four, but it's not the only way that might happen.
01:56
Maybe the first one is successful, then there are two failures, then a success.
02:00
That also has this probability, but fundamentally it's a different outcome.
02:06
We need to account for every possible order that these could come in.
02:10
This term tells us how many orders there are...