Two events are **dependent** if the outcome of one event affects the outcome of the other event.

For example, receiving a first place prize and receiving a second place prize in a race are dependent events. Once someone has won the first place prize, the **sample space** will decrease, so the probability of winning the second place prize will increase for those who are still in the race.

To find the probability of two dependent events, we multiply the probability of the first event by the probability of the second event.P\left(A\text{ and } B\right)=P\left(A\right)\cdot P\left(B\text{ after }A\right)

To understand this equation better, suppose you are selecting two cards from a deck of cards, one after the other. There are two ways this can happen:

Draw a card, then put it back in the deck before selecting another card

Draw a card and keep it, then select another card from the deck

The first method is described as "with replacement" because the card is placed back into the deck. The second method is called "without replacement" because the card is not placed back into the deck.

To decide which chores Jacques needs to do, he pulls out pieces of paper from a hat. The options are sweeping (S), mopping (M), or vacuuming (V).

Two chores on the same day

Two chores on different days

Which tree diagram shows the experiment with replacement?

What is the probability he will need to sweep and mop if they are done on the same day?

What is the probability he will need to sweep and mop if they are done on different days?

Is there a difference in the probabilities when selecting with or without replacement?

When selecting objects from a group "with replacement", selections are independent. Each time you select a card, you have the same probabilities.

When selecting objects from a group "without replacement", selections are dependent. Each time you select a card, you change the probabilities for the next selection.

This shows the probability of the second event is dependent on what color card was drawn first.

Esther was given four animal crackers as a snack. She has a donkey \left(D\right), an elephant \left(E\right), a goat \left(G\right), and a hippo \left(H\right). She eats one cracker and then another.

a

Are the selections independent or dependent?

Worked Solution

b

List the sample space for the possible two crackers she eats.

Worked Solution

c

Find the probability that Esther eats the elephant first, then the goat.

Worked Solution

A pile of playing cards has 4 diamonds and 3 hearts.

a

Find the probability of selecting two hearts if two cards are selected from the pile with replacement.

Worked Solution

b

Find the probability of selecting two hearts if two cards are selected from the pile without replacement.

Worked Solution

c

Is there a greater chance of selecting two hearts when the first card is replaced or when the first card is not replaced?

Worked Solution

Vanessa has 12 songs in a playlist. Four of the songs are her favorite. She selects shuffle and the songs start playing in random order. Shuffle ensures that each song is played once only until all songs in the playlist have been played. Find the probability that:

a

The first song is one of her favorites.

Worked Solution

b

Two of her favorite songs are the first to be played.

Worked Solution

Idea summary

Two events are dependent if the outcome of one event affects the outcome of the other event. The probability of two dependent events is given by:P\left(A\text{ and } B\right)=P\left(A\right)\cdot P\left(B\text{ after }A\right)

Events "with replacement" occur when the item drawn is placed back into the group before each selection. Each selection is independent of the others.

Events "without replacement" occur when the item remains outside of the group after selection. Each selection is dependent of the others. The probabilities of each selection will change depending on previous selections.