5.03 Probability of dependent events
Probability of dependent events
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.
Exploration
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).
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.
Examples
Example 1
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.
Are the selections independent or dependent?
List the sample space for the possible two crackers she eats.
Find the probability that Esther eats the elephant first, then the goat.
Example 2
A pile of playing cards has 4 diamonds and 3 hearts.
Find the probability of selecting two hearts if two cards are selected from the pile with replacement.
Find the probability of selecting two hearts if two cards are selected from the pile without replacement.
Is there a greater chance of selecting two hearts when the first card is replaced or when the first card is not replaced?
Example 3
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:
The first song is one of her favorites.
Two of her favorite songs are the first to be played.
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.