How to Calculate Simple Conditional Probabilities

How to Calculate Simple Conditional Probabilities

Conditional probability, just like it sounds, is a probability that happens on the condition of a previous event occurring. To calculate conditional probabilities, we must first consider the effects of the previous event on the current event.

What Is a Conditional Probability?

A conditional probability is a type of dependent event. Conditional probability involves finding the probability of an event occurring based on a previous event already taking place. To calculate a conditional probability, we must use the process of dependent events because the first event will affect the outcome of the second event. Let's review the topic of dependent events to help us better understand this process.

Dependent Events

Dependent events are events in which the previous attempts affect the outcome of subsequent events. Dependent events are just like they sound; each event is dependent upon what happened in the previous attempt. Let's look at an example of dependent events.

Walt has a big bag of gumballs. In his bag, he has 3 red, 6 green, 8 blue and 2 orange gumballs. What is the probability that Walt will reach into the bag and select a red gumball, then, WITHOUT REPLACING the gumball, reach into the bag and selecting another red gumball?

This probability would look like, P(red, without replacing and drawing another red).

We first need to find the probability of Walt selecting the first red gumball. By adding together all of the gumballs, we can see that there are 19 gumballs in the bag. There are 3 red gumballs in the bag, so the probability of getting a red gumball in the first draw is 3/19.

Next, we need to calculate the probability of Walt selecting a red gumball on his second draw. Remember, Walt did not replace the first gumball, and this will affect our totals. Walt now only has 2 red gumballs left in the bag, and the total number of gumballs is now only 18.

The probability that Walt's 2nd draw will be a red gumball is 2/18.

To calculate the probability of these two events occurring together, we would multiply the two events.

3/19 x 2/18 = 6/342

Remember that all answers must be in simplest form. 6/342 would be simplified to 1/57.

Walt has a 1/57 chance of drawing two consecutive red gumballs WITHOUT REPLACING the first gumball.

Conditional Probability

Remember, a conditional probability is a type of dependent event. Conditional probability involves finding the probability of an event occurring based on a previous event already taking place. The difference is that with conditional probabilities, we are just looking at the probability of one specific event occurring.

For example, thinking about Walt and his gumballs, Walt had 3 red, 6 green, 8 blue and 2 orange gumballs.

After first reaching in and selecting a red gumball, and WITHOUT REPLACING it, what is the probability that Walt's second draw will be another red gumball?

Walt knows that the probability of the first gumball being red is 3/19. Now, he has one less red gumball and one less total gumballs.

We can see that the probability of Walt's second gumball being another red would be 2/18. Remember, all fractions must be in simplest form. 2/18 would be simplified to 1/9.

The conditional probability that Walt's second gumball will be red after first drawing a red and not replacing it is 1/9.

Conditional Probability Example

Let's look at another example of conditional probability.

A group of teens are preparing to play a game of dodgeball. There are 30 teens that have arrived to play, 19 boys and 11 girls. What is the probability that the second player to be out is a boy after the first person out was also a boy?

We know that if the first person out is a boy, that there will only be 18 boys and 29 teens total left in the game for the 2nd round.

We can now see that the conditional probability that a boy is the second person to be out in dodgeball after the condition that a boy was the first out is 18/29. This fraction is in simplest form, so the conditional probability is 18/29.

Lesson Summary

Let's review the important facts that we need to remember when calculating a conditional probability. A conditional probability is a type of dependent event. Conditional probability involves finding the probability of an event occurring based on a previous event already taking place. To calculate a conditional probability, we must use the process of dependent events because the first event will affect the outcome of the second event. The difference is that with conditional probabilities, we are just looking for the probability of that one specific event occurring.