It happens every March. You lean forward to the edge of your chair and watch in horror, with a crumpled bracket sheet clutched in your hand as that desperation shot by that unheralded kid from that underdog school arcs slowly through the air. As it drops through the net and knocks out the team you picked to win it all you mutter “What are the odds?”
I understand the exasperation that missed predictions in the NCAA Tournament can bring, but I’m afraid that I don’t understand the exasperation of students who don’t understand basic probability. Basketball brings quite a few opportunities to illustrate the basics of probability, so let’s take a look at a few of those situations, so that you can get probability under control (even if your bracket stinks).
One of the areas of basketball most ripe for probability analysis is the free throw. Without the variables of defense or different areas of the floor, we can make a pretty good prediction about a player’s likelihood of making a free throw based on the results of his previous free throw attempts. For instance, if Player A has made 700 out of the 1000 free throws he has attempted, we would say that he is a 70% free throw shooter and that the probability of him making his next free throw is 0.70. It’s important to note that a probability is not a prediction. We don’t know whether Player A will make his next free throw, nor do we know that Player A would make 7 out of his next 10 free throws. All the probability does is reflect the likelihood of an individual event taking place.
As you might have noticed, probabilities are expressed as a decimal value between 0 and 1, where 0 is an event that will not occur and 1 is an event that is certain to occur. When we have multiple independent events and we want to know the probability of both occurring, we multiply the probabilities together.
Here’s an example. Player A gets fouled attempting a shot and is awarded two free throws. Seeing this, Chad says to Jimmy, “He’s a good free throw shooter, he’ll probably make both.”
“Probably not,” Jimmy replies.
Let’s take a look and see who is correct. First, we’ll take the probability of Player A making the first free throw, which we know is 0.7. Then, since the events are independent (the result of the first free throw doesn’t affect the likelihood of making the second free throw), we’ll multiply that by the probability of Player A making the second free throw.
=P(Making First)*P(Making Second) )
=0.70*0.70)
=0.49)
So, Jimmy is correct. Player A will make both free throws less than half of the time, even though he is likely to make any single free throw attempt. Just as Jimmy finishes bragging and Chad goes back to his drink, Player A is fouled shooting a 3-pointer and is awarded three free throw attempts.
“I’ve got it this time,” Chad says. “I know he’s going to make at least two of these free throws almost all time time.”
“Don’t count on it,” Jimmy says.
To figure out how often Player A will make at least two free throws, we first need to figure out how many different ways that could happen. Here is a table that summarizes that information:
| |
Attempt 1 |
Attempt 2 |
Attempt 3 |
| Outcome 1 |
Make |
Make |
Make |
| Outcome 2 |
Make |
Make |
Miss |
| Outcome 3 |
Make |
Miss |
Make |
| Outcome 4 |
Make |
Miss |
Miss |
| Outcome 5 |
Miss |
Make |
Make |
| Outcome 6 |
Miss |
Make |
Miss |
| Outcome 7 |
Miss |
Miss |
Make |
| Outcome 8 |
Miss |
Miss |
Miss |
So, in 4 of 8 of the sequences we could have Player A makes at least two shots. However, we’re not done yet because not all of these outcomes are equally likely. Player A is a pretty good free throw shooter, so he’s more likely to make all three than to miss all three. So, we need to find the probability of each of the four different scenarios we want and add them together. We add in this “or” situation because either Outcome 1 or Outcome 2 or Outcome 3 or Outcome 5 will satisfy the condition we’ve set.
=0.7*0.7*0.7=0.343)
=0.7*0.7*0.3=0.147)
=0.7*0.3*0.7=0.147)
=0.3*0.7*0.7=0.147)
=P(O1)+P(O2)+P(O3)+P(O5))
=0.343+0.147+0.147+0.147=0.784)
Is that the outcome you expected? Whether it’s intuitive or not, working with probabilities will help you get more comfortable with them on your test… even if it doesn’t help your bracket.
Good luck with probabilities and with your picks!
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