Video Transcript
In an experiment in which a fair coin is tossed five consecutive times, let 𝑥 be the discrete random variable expressing the number of heads minus the number of tails. Find the probability distribution of 𝑥.
The probability distribution of a discrete random variable is a list or table containing all the possible values the discrete random variable can take together with their associated probabilities. So, we must first determine all the possible values that 𝑥 can take and then calculate their probabilities. The question states that this fair coin is flipped five times and 𝑥 is the discrete random variable that represents the number of heads minus the number of tails. So, let’s consider what could happen.
Well, all five flips of the coin could result in heads, in which case there would be no tails and the value of 𝑥 would be five minus zero, which is five. Or four flips of the coin could result in heads, in which case there would be one tail and the value of 𝑥 would be three. There could be three heads and two tails, in which case the value of 𝑥 would be one, or two heads and three tails, in which case the value of 𝑥 would be negative one; it’s two minus three. There could be one head and four tails, in which case the value of 𝑥 would be negative three. Or finally, all five flips could result in tails, in which case there are no heads and the value of 𝑥 is negative five.
So, the six values that this discrete random variable can take are five, three, one, negative one, negative three, and negative five. We denote these as 𝑥 subscript 𝑖, and we write them in the top row of our table. In the second row of the table, we’re going to write the corresponding probabilities, which we denote as 𝑓 of 𝑥 sub 𝑖. Now, to work out these probabilities, we need to count up how many different ways there are of each of these different outcomes occurring.
For some of the outcomes, this is very straightforward. There’s only one way that we can get all heads and only one way that we can get all tails. For others though, we might have to think a little bit harder. In the case of getting four heads and one tail, there are five possible positions that the one tail could occur in. So, there are five different ways of this occurring. By symmetry, the same is true for the number of different ways of getting one head and four tails because we can just swap all the heads for tails and all the tails for heads.
For three heads and two tails, we have to think even harder. But if we list all of the possible outcomes systematically, we see that there are in fact 10 different ways of getting three heads and two tails. By symmetry, there are also 10 different ways of getting two heads and three tails because we can swap all the heads and tails around. In fact, if you’re familiar with combinations, then this would be a far more efficient method of working these numbers out. For example, if you want to know the number of ways of getting three heads, then this is equivalent to five choose three because we’re choosing three of the five coin flips to be heads.
Now that we found the number of ways that each outcome can occur, we can find the total number of ways by summing these values, and it is 32. The probability for each value of 𝑥 will then be the number of ways that outcome can occur over the total of 32. So, for example, for the value of 𝑥 of five, there’s one way that can occur when we get all heads and no tails. So, the probability is one over 32. We can then fill in all of the remaining probabilities. And even though some of these fractions can be simplified, we’ll keep them all with a common denominator of 32. We may also prefer, and this is just convention, to write the values in the probability distribution in ascending order.
So swapping the values around, which is quite easy to do here because the distribution is symmetric, we have the probability distribution of 𝑥. The values 𝑥 can take are negative five, negative three, negative one, one, three, and five. And the corresponding probabilities are one over 32, five over 32, 10 over 32, 10 over 32, five over 32, and one over 32.
