PROBABILITY DISTRIBUTIONS DISCRETE DISTRIBUTIONS BINOMIAL POISSON HYPERGEOMETRIC NEGATIVE

PROBABILITY DISTRIBUTIONS DISCRETE DISTRIBUTIONS BINOMIAL POISSON HYPERGEOMETRIC NEGATIVE BINOMIAL

LEFT OR THE RIGHT? WIN 100 POINTS WIN 300 POINTS

WHICH IS WHICH? WIN 100 POINTS WIN 300 POINTS

RANDOM VARIABLES A random variable is a function that assigns a real number to every sample point of the sample space. § Discrete – assumes a finite number of possible values § Continuous – assumes infinitely many possible values Probability Distribution – table, graph, or formula assigning a probability for each possible value of the random variable For now, let us consider DISCRETE PROBABILITY DISTRIBUTIONS PLBautista 4

EXPECTED VALUES PLBautista 5

EXAMPLE If you bet 100 points under (K or Heart), are you expected to win or the lose in the long run? § Recall that you get 300 points if you make a correct bet under (K or heart) How about if you bet under the (7 or 11)? What if you bet under (even)? § Recall that you get 300 points if you make a correct bet under (7 or 11) PLBautista 6

EXAMPLES Suppose that in a game of tossing two coins, you win Ph. P 100. 00 when two heads come out and you lose Php 50. 00 otherwise. What is your expected gain? Suppose that in a game, you win Ph. P 50. 00 when you draw a heart card and you lose Ph. P 30. 00 otherwise. What is your expected gain? PLBautista 7

EXAMPLES In a gambling game, a man is paid P 250 if he gets all heads or all tails when three coins are tossed, and he will pay out P 150 if either one or two heads show. What is his expected gain? Suppose that an antique jewelry dealer is interested in purchasing a gold necklace for which the probabilities are 0. 22, 0. 36, 0. 28, and 0. 14, respectively, that she will be able to sell it for a profit of P 1, 250, sell it for a profit of P 750, break even, or sell it for a loss of P 750. What is her expected gain? PLBautista 8

SPECIAL DISCRETE DISTRIBUTIONS PLBautista 9

BINOMIAL DISTRIBUTION PLBautista 10

EXAMPLE In a survey, it was found that 33% of students aged 16 -22 own a credit card. In a sample of 12 students, what is the probability that § Exactly one has a credit card? § Exactly 4 have credit cards? § At least one has a credit card? § What is the expected number of students with credit cards? PLBautista 11

POISSON DISTRIBUTION PLBautista 12

EXAMPLE A secretary makes two errors per page, on the average. What is the probability that on the next page, she will make § No error? § Four or more errors? PLBautista 13

HYPERGEOMETRIC DISTRIBUTION PLBautista 14

EXAMPLE Of 25 students (14 boys and 11 girls), 5 were absent on Thursday. What is the probability that § Two of the absentees were girls? § Two of the absentees were boys? § At least one absentee is a boy? PLBautista 15

NEGATIVE BINOMIAL DISTRIBUTION PLBautista 16

EXAMPLE (SLLL, 2013) Rechel, a bank teller, knows from experience that 40% of customers entering the bank will make a cash withdrawal. What is the probability that on any given banking day, the tenth customer to enter the bank will be § The third customer to make a cash withdrawal? § The first customer to make a cash withdrawal? PLBautista 17

EXERCISE Five percent of truck drivers are women. If ten truck drivers are chosen at random, what is the § Expected number of women? § Probability that two are women? § Probability that at least one is a woman? Phone calls arrive at the rate of 48 per hour. What is the probability that 3 calls arrive in a 5 -minute span? PLBautista 18

EXERCISE (SLLL, 2013) Pocholo has four pairs of black socks and two pairs of white socks in his bedroom drawer. If he randomly picks two socks from his drawer, § Provide the probability distribution table for the number of black socks selected. § Solve for the probability that he will get exactly one pair of black socks. (SLLL, 2013) Teacher Meldy gave her statistics class a homework to be discussed in the following class meeting. If only 45% of her class diligently do their homework, what is the probability that during the next class meeting, the seventh student selected at random will be § The third student called who did the homework? § The first student called who did the homework? PLBautista 19