A discrete probability distribution is one in which the outcomes of an event are separate and distinct. A good example is the roll of a die: while it is possible to roll a two or a three, it is impossible to roll a 2.5.
The opposite of a discrete distribution is a continuous distribution. An example of a continuous distribution would be the distribution of heights of all New Yorkers. It is possible for a New Yorker to be 70 inches tall, or to be 71 inches tall, or to be any height in between. Someone might be 70.2 inches tall, 70.25 inches tall, or 70.25478 inches tall (assuming you have a very precise measuring stick!) The point is, a height could theoretically take any value between 71 and 72 inches. As we’ve already seen, this is not the case with rolling a die.
This article will focus exclusively on understanding discrete probability distributions. For more about the continuous distribution, and the differences between discrete and continuous, check out this previous article of mine.
Discrete probability distribution of four coin flips
Let’s consider the possible outcomes of four consecutive coin flips. They are listed in the following table. The probability of any one particular outcome (HTHT, TTHH, etc.) is always (½)4 (see this article on independent events if you’re unsure why). We then multiply this probability by the total number of possible combinations for a given outcome to get the following:
|3 heads, 2 tails||HHHT, HHTH, HTHH, THHH||4×(½)4=0.25|
|2 heads, tails||HHTT, HTHT, HTTH, TTHH, THTH, THHT||6×(½)4=0.375|
|1 heads, 3 tails||HTTT, THTT, TTHT, TTTH||4×(½)4=0.25|
The table above is a discrete probability distribution. However, to illustrate it a bit better, let’s graph the probabilities to give a visual sense of the distribution.
We can see the discrete nature of this distribution visually because there is a jump from the height of one bar to the next. This is because, as we know, it’s impossible to toss the coin four times and get 2.5 heads and 1.5 tails. The outcomes shown above are exhaustive, meaning they form a complete list of all the possible outcomes of four coin tosses.
What happens to the distribution if we toss the coin 50 times? We end up with the following distribution:
Note that as long as the number of coin flips is finite, we will have a discrete probability distribution, with jumps from bar to bar. But, for those of you who are familiar with the normal distribution, you can see that as the number of coin flips gets larger, the distribution above is approaching the normal bell curve.
Next, let’s look at an example and find the expected value of a discrete probability distribution.
Expected value of discrete probability distribution
John is a student in an introductory statistics class. He has an upcoming exam, and the only possible grades on the exam are multiples of 10: 10, 20, 30, etc., all the way up to 100. Let’s assume that the discrete probability distribution for the grade John will get on his next exam looks as follows. (We are assuming the probability he gets below a 60 is zero.)
Expressing this graphically, we have:
From this probability distribution, we can now answer the question: what is John’s expected score on the next exam? We might guess that he should get an 80, since this has the highest probability of occurring. However, this would be incorrect, as the expected value of the distribution will be a weighted average of all possible outcomes:
E(X) = 60×0.1 + 70×0.25 + 80×0.4 + 90×0.2 + 100×0.05 = 78.5
Therefore, John’s expected score is 78.5—slightly less than an 80. This is because he has a greater probability of scoring 60 and 70 than he does of scoring 90 or 100. This skews the expected value towards the left end of the distribution.
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