2. Laws of Probability

4. What is the probability of throwing a pair of dice and obtaining a 5 or a 7? These are mutually exclusive events. You can’t throw a pair of dice and obtain 5 and 7. So, What is the probability of throwing a pair of dice and obtaining a 5 or and odd number? These are not mutually exclusive conditions. 5 is an odd number, so P(5 and 7) is not equal to zero. Therefore, we must subtract the probability of obtaining a 5 and an odd number. So,

6. A deck of playing cards has 52 cards. Half of these are red and half are black. Half of the red cards are hearts and half are diamonds. Half of the back cards of clubs and the other half are clubs. This gives us four “suits”, two red and two black. Each suit has cards numbered from one (“ace”) to ten and there three “royals” (Jack, Queen, and King) in each suite. This gives us four suits of 13 cards each for a total of 52 cards. If we choose a card, read it, replace it into the deck and reshuffle the cards before drawing a second card, what it is probability that both cards were red cars? We know that the probability of two events occurring at the same time is the product of their probabilities, so

7. What if we do not replace the first card that we drew? Now what would the probability of drawing two red cards be. The answer to this is a big “It depends.” What it depends on is whether the first card were a red card or a black card since that would determine how many black and red cards would be in the deck when the second card was drawn. There would be 51 red cards left if the first card drawn were red, but there would be 52 left if the first card drawn were black. These two events (the choosing of cards) are not independent. So, P(red|red) is 25/51 since if the first card drawn were red, there would be 51 cards left in the deck and only 25 would be red. So, P(red|red)=25/51=.490.

8. Combinations Is the number of ways N things can be chosen n at a time.

9. Ten assistant professors have applied for tenure. There are five men and five women applying and it turns out that 4 women and 2 men are awarded tenure. If they were tenured randomly, what is the chance that men would have done as well or even less well than they did? Since we first need to know the total number of combinations of six faculty, men and women, out of the 5 men and 5 women who could get tenure. This is found by generating samples of 10 people taken 6 at a time. So, Possible gender combinations when there are 6 people in this case are: 5 men, 1 woman 4 men, 2 women 3 men, 3 women 2 men, 4 women 1 man, 5 women

10. To determine the number of outcomes that have 6 people tenured, remembering that the selections of men and women are independent, we find n mf for all possible combinations of men and women by: So, the probability of having 2 men and 4 women obtain tenure, just by chance is