Exercise 9 Page 717

We can illustrate the difference between dependent and independent events by using two variations of the same scenario. We want to choose two cards from a standard deck, once without replacement and once with replacement.

Choosing a Card Without Replacement

In this scenario, we will draw one card and then, without replacement, we will draw another card. We want to know the probability of drawing a 3 of diamonds and then a 9 of clubs.

We know there are 52 cards in a deck. We can write the probability of drawing a 3 of diamonds. P(3of Diamonds) = 1/52 Since we are not replacing the card, there are 51 cards remaining in the deck. Now we can write the probability of drawing a 9 of clubs given that we already have a 3 of diamonds. P(9of Clubs| 3of Diamonds) = 1/51 Using these probabilities, we can find the probability of drawing a 3 of diamonds and then a 9 of clubs.

Choosing a Card With Replacement

Now consider that we replace the card that we drew before drawing the second card. The probability of drawing a 3 of diamonds first remains the same. P(3of Diamonds) = 1/52 Since we are replacing the card in the deck, for our second card, the deck has 52 cards again. Therefore, the probability of drawing a 9 of clubs given that we drew a 3 of diamonds is different than it previously was. P(9of Clubs | 3of Diamonds) = P(9of Clubs) ⇓ P(9of Clubs | 3of Diamonds) = 1/52 Since P(9of Clubs | 3of Diamonds) is the same as P(9of Clubs), these events are independent. Now we can find the probability of drawing a 3 of diamonds and then a 9 of clubs.