00:01
It's given here that based on historical data, the probability that a major league pitcher in a single game pitches a no -hitter is about 1 out of 13 ,000.
00:10
So that's a probability of 1 over 1 ,300.
00:15
And then we consider 650 games, and we want to find the probability that we have 0, 1, 2, or 3 no -hitters.
00:24
So let's first define a random variable x as the number of no -hitters.
00:28
We're asked to use the binomial distribution to solve this, and the probability function for the binomial random variable is given by this formula.
00:52
So for the first one, we want to find the probability of zero no -hitters, that's the probability that x is equal to zero.
00:59
In this case, the probability function simplifies to 1 minus p, which is 1299 over 1 ,300, to the exponent n, which is 650.
01:13
So it would be 650.
01:18
And this comes out to probability 0 .60 and 64.
01:24
And if we do the same thing for x equals 1, the probability mass function gives us 650 choose 1 times 1 over 1 ,000 to the exponent 1 times 1299 over 1 to the exponent 649.
01:48
This comes out to probability of 0 .3034.
01:54
Now this is bit tedious but not too bad to do all 4 of the 1.
01:58
These by hand.
02:00
So let's use software for the last two.
02:03
So we have the probability that x is equal to 2, probability that x is equal to 3.
02:09
So if we go to excel, i'll show you how to do all of them all in once.
02:15
So let's first, in the first column, but all of the values that x can take on, so that's, we want to solve the probabilities for x equals 0 through 3.
02:27
In column b, we're going to use the binomial distribution function.
02:31
So we start with an equal cosine, use the binomial distribution function, so we select that.
02:37
For the number of successes, i select the cell in column a, then i hit a comma, and for the number of trials, i enter 650, probability of success.
02:50
That's 1 over 1 ,300, which is approximately .007692.
02:57
And then for the cumulative argument, we enter true because we want the probability that that x is exactly equal to the value in column a.
03:09
In this case, that's 0.
03:11
Hit enter.
03:13
And so we get the same value that we calculated for the probability of x being equal to 0.
03:18
We can now drag this formula down for all possible values of x...