Sometimes probabilities need to be calculated when more than one event occurs. These types of compound events are called independent and dependent events. Through this lesson, we will look at some real-world examples of how to calculate these probabilities.

## Independent and Dependent Events

**Independent events** are events that do not affect the outcome of subsequent events.

In an independent event, each situation is separate from previous events. An example of an independent event would be selecting a card from a deck of cards and then returning the card to the deck. After you return the card, select another card from the same, equal deck.

**Dependent events** are just the opposite. 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. An example of a dependent event would be selecting a card from a deck of cards and not replacing the card.

Then you draw another card from the now-smaller deck of cards.

## Probability

The **probability** of an event occurring is a ratio that states the likelihood of an event happening. To find the probability of a single event, you will use the ratio of the number of favorable outcomes over the number of total outcomes.Let’s look at an example: What is the probability of selecting a king from a standard deck of cards?Looking at the deck of cards, we know that there are 52 cards in each deck. The number of total outcomes is 52 because we could select any of the 52 cards when choosing at random. We can also see that there are 4 kings in each deck of cards.

So, the number of favorable outcomes would be 4.We can see that there is a ratio of 4 to 52 chances of selecting a king at random from a deck of cards. All ratios must be in simplest form, though, so 4/52 will be reduced to 1/13. The probability of selecting a king at random from a standard deck of cards is 1/13.Sometimes you will be asked to find the probability of more than one event occurring in consecutive order.

When this occurs, you will need to multiply both probabilities together to calculate the combined probability. Let’s look at an example where you will be asked to find the probability of more than one event occurring: What is the probability of rolling a standard die and getting a 2 and then rolling again and getting another 2?To start this problem, you will need to calculate the probability of each event happening independently. The probability of rolling a 2 on a standard die can be found by using the formula: total number of favorable outcomes over the total number of possible outcomes. On a die, there are 6 total outcomes, and only one of the outcomes is a 2. So, the number of favorable outcomes is 1 and the number of total outcomes is 6. So, the probability of rolling a 2 is 1/6.Since the probability of rolling a 2 is 1/6, the probability of rolling a 2 on the next roll would be the same.

To calculate the probability of both events happening together, we will need to multiply the two probabilities together. By multiplying these two probabilities together, you get 1/36. So, the probability of rolling a die and getting a 2, then rolling a die again and getting another 2, would be 1/36.

## Probability of Independent Events

Again, **independent events** are the events that do not affect the outcome of subsequent events. The last situation was an example of an independent event. The rolling of the die and getting a 2 did not affect the outcome of the second event of rolling the die again.

Each rolling of the die is an independent event.Let’s look at another example of an independent event. Jamie and Sam were having a debate and decided the best way to settle their dispute was by flipping a coin. Jamie decided to be heads, and Sam decided to take tails. Jamie is now curious: What is the probability that by flipping a coin 2 times they will receive a heads each time?To find this probability, Jamie must find out the probability of each event occurring separately.

To calculate the probability, Jamie must use the formula: the number of favorable outcomes over the number of possible outcomes.Jamie knows that the coin has two sides, and only one of them is heads. So, the probability of Jamie getting a heads on the first flip is 1/2. As the boys prepare to flip a coin for the second time, they know the probability of flipping a coin and getting a heads is 1/2. The probability of getting a heads on the second flip is also 1/2. To find the probability of these two events happening together, we need to multiply these two probabilities together. When we multiply 1/2 times 1/2, we get 1/4.

Jamie now knows that the probability of flipping a coin twice and getting heads both times is 1/4.

## Probability of Dependent Events

Just the opposite of independent events, **dependent events** are events in which 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 these dependent events.James the Superb Magician likes to dazzle and amaze his audience with a card trick in which he selects two cards at random from a deck of cards but announces the cards that he will select prior to selecting them.

James asks for an audience member to join him for the trick. Wendy, a female audience member, raises her hand to volunteer. The crowd gives her a rousing applause as she makes her way to the stage. James explains to Wendy that he will select a card at random from the deck of cards. Wendy examines the deck of cards to make sure the deck is fair. James announces that he will first draw an ace from the deck.Wendy wonders: What is the probability that James will select an ace from the deck of cards? She knows that there are 4 aces out of 52 cards.

James explains to Wendy that he will now select another card from the deck, and it will also be an ace. Wendy is perplexed at how James could be lucky enough to draw two aces in a row from the deck of cards. She thinks to herself: What is the probability of James selecting an ace and then, without replacing the card, selecting another ace?Wendy knows that to find the probability of him selecting an ace from the deck of cards, she will need to use the formula: total number of favorable outcomes over the total number of outcomes. In the first event, the number of favorable outcomes is 4 because James was selecting one of the 4 aces. The total number of outcomes is 52 because there are 52 cards in a standard deck of cards.

So, Wendy knows the probability of James selecting an ace on the first draw is 4/52.James turns over the first card that he selected to show it, in fact, was an ace. The crowd cheers with excitement.Wendy now must calculate the probability of James getting an ace on the second draw. She knows that James did not replace the card, so there are only 51 cards left in the deck. She also knows that there could be only 3 aces left because the first card he selected was an ace. So, the probability of him getting an ace on the second draw is 3/51.

To find the probability of James getting an ace on the first card and then, without replacing it, getting an ace on the second card, Wendy needs to multiply these two events together. She will need to multiply 4/52 x 3/51. By multiplying these two probabilities together, she gets 12/2652. She reduces the fraction to 1/221. The probability of James selecting an ace from a deck of cards and then, without replacing the card, selecting another ace is 1/221.

## Dependent Events Example

Let’s look at another example: What is the probability of selecting a spade from a standard deck of cards and then, not replacing the card, selecting an ace? By looking at this question, we know that this is a dependent event because the card is not replaced.

Therefore, the first event will affect the results of the second event. To calculate the probability of both of these events occurring, we will need to find the probability of each event separately and then multiply the two probabilities together.To calculate the probability of the first event – selecting a spade from a standard deck of cards – we will use the formula: total number of favorable outcomes over the total number of outcomes.

In a standard deck of cards, there are 4 different suits, each containing 13 cards. So, there are 13 spades in each deck of cards, which would be the number of favorable outcomes. The total number of outcomes would be the same as the number of cards in the deck, which is 52. The probability of selecting a spade from a standard deck of cards is 13/52. To make this problem easier, though, we can reduce the probability 13/52 to 1/4.

Now let’s find the probability of the second event, selecting an ace from a standard deck of cards, occurring. Remember that after the first event, the card was not replaced. This affects the number of cards that are now in our deck of cards.

With one card selected, there are only 51 cards remaining in our deck of cards. So, the total number of outcomes left is 51. As far as the number of favorable outcomes, one spade was selected in the first event. The card could have been the ace of spades.

Since this card was discarded, the number of favorable outcomes that we know for certain remain in our deck of cards is 3. We know that the probability of the second event occurring is 3/51. Again, to make this problem easier, we can reduce this probability to 1/17.To calculate the probability of both events occurring in connection with each other, we need to multiply both probabilities together.

So, the first event of selecting a spade and then not replacing it was 1/4, and the probability of selecting an ace as the second event was 1/17. By multiplying these two probabilities together, we get 1/68 as the probability of both of these events occurring together.

## Lesson Summary

In this lesson, we’ve looked at two types of probabilities: independent and dependent events. **Independent events** are events that do not affect the outcome of subsequent events.

To calculate the probability of an independent event, we use the formula: total number of favorable outcomes over the total number of outcomes.**Dependent events** are just the opposite. Dependent events are events in which the previous attempts affect the outcome of subsequent events. The key difference is where the second event is affected by the first event. To calculate the probability of each of these events, we will use the same formula as we did for the independent event, but once you find the probability for each event, we multiply them together to find the probability of both events occurring together.

Remember that to make the problem easier, we can reduce the probability, if possible.

## Learning Outcomes

Following your completion of this lesson, you might be able to:

- Differentiate between dependent and independent events
- Calculate the probabilities of both dependent and independent events occurring