00:01
So this question is asking us to construct the probability distribution function for the number of heads gotten when a fair coin is flipped three times.
00:12
So the probability distribution function basically is a function of x that evaluates to the probability that x equals a certain value at x.
00:28
And it has to follow the rules that f of x is always greater than are equal to zero, and the sum across all x of f of x equals one.
00:44
So that being said, now let's look at the probabilities associated with getting a certain number of heads when flipping a fair coin three times.
00:52
And so to do this, we can construct a probability tree because each coin flip is independent of the other, each probability for heads or tails is 0 .5.
01:07
Right, and each one is independent of the other, so we can continue this out, heads or tails, sorry, that shouldn't be there.
01:18
0 .5, 0 .5 heads, tails.
01:22
Do it one more time because we're flipping it three times.
01:46
All right, and each of these probabilities is 0 .5.
01:49
So when we flip the coin three times, there's a total of eight, one, two, three, four, five, six, seven, eight, distinct outcomes, each with an equal probability of 0 .5 cubed, or one -eighth, or point -125.
02:09
All right, so the probability of this is 0 .125, the probability of this is 0 .125, etc.
02:15
And so now to construct our probability distribution function, we need to figure out the probability of getting a certain number of heads for each possible value.
02:31
So first, what is the probability that the number of heads we get is equal to zero? well, there's only one outcome in which the number of heads equals zero, and that's when we get all tails.
02:42
And so i'm going to go and actually write the total outcome here, just so we can see things a bit more explicitly.
03:04
And you can see here that there's only one outcome that is all tails, and so this is going to be equal to 0 .125.
03:15
The probability that there is one head is equal to the number of outcomes divided by 8.
03:22
So here we have one outcome, two outcome, three outcomes.
03:26
Yeah, there's only three outcomes in which there's only one head.
03:34
So this is equal to 3 over 8, or 0 .375...