1. Refreshing Your Skills for Chapter 10

2.  If you flip a coin, the probability that it lands with heads up is 1/2.  If you roll a standard die, the probability that it lands with either the 1 or the 3 up is 2/6.  In each fraction, ◦ The denominator is the number of possible outcomes, or results, that are equally likely. ◦ The numerator is the number of those outcomes that are desirable.  In this lesson you will review basic probability to prepare for an in-depth look at probability in the rest of the chapter.

3.  If you roll a pair of standard dice, what is the probability that they land with a sum of 5 showing?

4.  You need to determine the number of possible equally likely outcomes and the number of desirable outcomes for this experiment. You might think that there are 11 possible sums of two dice (2, 3, 4, 5, 6, 7, 8, 9, 10, 11and 12), so the probability of a 5 would be 1/11.  However, the 11 sums are not equally likely. One way to see this is to list all of the possible rolls in an organized fashion. In the table you can see there are 36 possible equally likely outcomes. The four pairs that sum to 5 are highlighted. So the probability that the sum of the two dice is 5 is 4/36, or 1/9.

5.  If you roll three dice, what is the probability that they sum to 5?

6.  In the table for Example A, each of the six numbers on the first die is paired with the six numbers on the second die. So the total number of possible outcomes is 6 x 6, or 36. If you include a third die, then each of these 36 two dice outcomes can be matched with each of the six numbers on the third die.

7.  So the total number of possible outcomes is 6 x 6 x 6, or 216. How many of the outcomes sum to 5?  Make an organized list: ◦ (1, 1, 3), (1, 2, 2), (1, 3, 1) The possibilities with a 1 on the first die. ◦ (2, 1, 2), (2, 2, 1) The possibilities with a 2 on the first die. ◦ (3, 1, 1) The only possibility with a 3 on the first die.  There are six desirable outcomes, so the probability of three dice summing to 5 is 6 out of 216 or 6/216, or 1/36.

8.  If you choose one card at random from a standard deck, what is the probability that the card you choose is from the hearts suit? A standard deck of cards contains 52 cards. There are four different suits: clubs ♣, diamonds♦, hearts♥, and spades♠. Each suit contains 13 cards labeled ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king. Two suits are red: diamonds and hearts. The other two suits are black: clubs and spades. If you want to get a heart, you’re interested in 13 outcomes out of the 52 possible outcomes. So the probability that your card is a heart is 13/52, or 1/4. (1/4 of the cards are hearts♥.)