COUNTING RULES PROBLEMS 1. How many different ways can a nurse visit 9 patients if she wants to visit them all in one day? If she wants to visit only 5? 2. How many different 4-digit ID tags can be made if the digits can be used more than once? If the first digit must be 5 and repetitions are not permitted? 3. If there are 4 major roads from city X to city Y and 7 major roads from city Y to city Z, how many different trips can be made from city X to city Z passing through city Y? 4. How many different selections of 3 presents can be made if there are 10 presents under a Christmas tree? 5. How many different selections of 12 cards can be made from a standard deck of cards? 6. How many ways can 4 baseball players and 3 basketball players be selected from a 12 baseball and 9 basketball players?

COUNTING RULES AND PROBABILITY PROBLEMS 1. What is the probability of getting 4 aces when 4 cards are drawn from an ordinary deck? When 7 cards are drawn? 2. A box contains 24 transistors of which 4 are defective. If 6 transistors are sold, find the probability that exactly 2 defectives are sold? No defectives are sold? All defectives are sold? 3. A combination lock consists of the 26 letters of the alphabet whose key is a 3-letter code. Find the probability that ABC will unlock the device? 4. Four balls are taken in succession from a box containing 7 balls labelled 2, 3, 5, 6, 7, 8, 9. What is the probability that the 4-digit no. obtained is an even no? That the 4-digit no. obtained is less than 700? That the first ball taken is even?

COMPOUND EVENTS (JOINED BY ‘OR’) Example: Consider the selecting a card from a standard deck. a. A = Getting a spade? b. B = Getting a 7? P(A) = 12/52 P(B) = 4/52 c. C = Getting a king? P(B and C) = 0/52 = 0 d. B or C = Getting a 7 or a king?P(B or C) = 8/52 Note: P(B or C) = 8/52 = P(B) + P(C) – P(B and C) d. B and C = Getting a 7 king? Note: ‘and’ means ‘both, at the same time’ Event B and Event C has no common outcome. They are said to be MUTUALLY EXCLUSIVE. The prob. of mutually exclusive events is 0.

e. A and B = Getting a 7 spades? Event A and Event B are not mutually exclusive. P(A and B) = 1/52 f. A or B = Getting a 7 or spades?P(A or B) = 15/52 Note: P(A or B) = 15/52 = P(A) + P(B) – P(A and B) The probability of a compound event joined by ‘OR’ is: P(A or B) = P(A) + P(B) – P(A and B) The probability of a compound event joined by ‘OR’ is: P(A or B) = P(A) + P(B) – P(A and B) Example: The probability that a given tourist goes to the amusement park is 47% and the probability that he goes to the water park is 58%. The probability that he goes either to the amusement or to the water park is 95%. What is the probability that he visits both parks?

Example: A statistics class consists of 18 juniors and 10 seniors; 6 of the seniors are females, and 12 of the juniors are males. If a student is selected at random, what is the probability of: a. Selecting a junior or a female? b. Selecting a senior or a male? c. Selecting a junior or a senior? Example: The table below gives the college degrees awarded in a the recent academic year: Bachelor’sMaster’sDoctorate Men570,000210,00025,000 Women470,000300,00021,000 If a degree holder is chosen, find the probability of getting: a. A bachelor’s degree holder? c. A doctorate degree holder or a woman b. A doctorate degree awarded to a woman

COMPOUND EVENTS (JOINED BY ‘AND’) Example: A card is selected from a deck, re-placed (placed back in the deck) and then a second card is drawn. Find the prob. of a. A = Selecting a queen?P(A) = 4/52 b. B = Selecting an ace (after Event A)?P(B|A) = 4/52 c. A and B = Selecting a queen and then an ace? S.S. = {A ♣- A ♣, A ♣- 2 ♣, …, K ♥ -Q ♦ } A and B = {Q ♣- A ♣, …, Q ♥ -A ♦ } 52x52 = x4 = 16 P(A and B) = 16/2704 Note: P(A and B) = P(A) x P(B|A) In this experiment, the chance of A and the chance of B has no effect on each other. These events are said to be INDEPENDENT. X|Y means Event X, after Event Y has been done or Event X, given Event Y has occurred X|Y means Event X, after Event Y has been done or Event X, given Event Y has occurred

Example: A card is selected from a deck, not re-placed and then a second card is drawn. Find the prob. of a. A = Selecting a queen?P(A) = 4/52 b. B = Selecting an ace?P(B) = 4/51 c. A and B = Selecting a queen and then an ace? S.S. = {A ♣- 2 ♣, A ♣- 2 ♣, …, K ♥ -Q ♦ } A and B = {Q ♣- A ♣, …, Q ♥ -A ♦ } 52x51 = x4 = 16 P(A and B) = 16/2652 In this experiment, after Event A has happened, the chances of Event B is affected. Events A and B are therefore dependent. Note: P(A and B) = P(A) x P(B|A)

The probability of a compound event joined by ‘AND’ is: P(A and B) = P(A) x P(B|A) The probability of a compound event joined by ‘AND’ is: P(A and B) = P(A) x P(B|A) Example: Manulife found that 53% of the residents of a city has a homeowner’s insurance with them; and, of these clients, 27% also had automobile insurance with them. Find the probability that a resident of this city has both homeowner’s and automobile insurance with Manulife. Example: A recent survey asked if people thought women in the armed forces should be allowed to participate in combat. GenderYesNo Male3218 Female842 Find the probability that a. the respondent is a female who agrees b. the respondent is a male who disagrees