Finite Final
Parts of a probability distribution table
xi = item fi/n = relative frequency fi/n x xi xi - u (item - mean) (xi-u)^2 fi/n x (xi-u)^2 (add all these up to get the variance)
Gambling: $1 bet for any one of 38 numbers on a roulette wheel. Bettor wins $36 (which includes original bet) if the ball lands on the selected one slot, otherwise they win no money and do not get the $1 back What is the probability for the earnings of this bet and the expected value?
1. 1/38 = $36 // 37/38 = $0 E(X) = 0 (37/38) + 36 (1/38) = 18/19 Answer: On average, one would earn 18/19's of a dollar
Committee of senators question: If there are 6 senators making up a committee and there are 100 senators in total (2 from all 50 states), how many ways can you make this committee without having two senators from the same state?
1. C(50,6) 2. 2^6 3. C(50,6) x 2^6
Defective fuses question: fuses packaged in boxes of 8, three fuses are selected for inspection, box is rejected if at least one of the three are defective, probability that box containing four defective fuses will be rejected?
1. C(8,3) = 56 2. Find C(4,1) x C(4,2)//C(4,2) X C(4,1)//C(4,3) x C(4,0) 3. 24 + 24 + 4 = 52 4. 52/56 = 13/14 Answer is 13/14 **Remember to use three for both finding the total and for each individual case**
Dice: A pair of dice is rolled 720 times. Find the expected number of times a sum of 7 will occur.
1. Calculate the amount of times a 7 can occur >> 6/36 which simplifies to 1/6 2. 720(1/6) = 120 times
Gambling: What is the average number of clubs in a poker hand? (There are 52 cards in a deck, 5 cards in a poker hand, and 13/52 are clubs).
1. Construct a probability distribution table Total = (52 5) >> 52-13 = 39 (all cards that are not clubs) 0 clubs = freq. (39 5) >> rel. freq. (39 5) / (52 5) 1 club = freq. (39 4) x (13 1) >> rel .freq. (39 4) x (13 1) / (52 5) ... Add together all of the results from 0-5 and then you have an answer
Grid question: How many ways to get from A to B through C?
1. Count how many blocks it takes to get from A to C, then do C(total number, # of blocks down) 2. Same thing as step 1 for C to B 3. Multiply both totals for the answer
A sample of two balls is drawn from an urn containing 4 white balls and 5 red balls. Are the events "the sample contains at least one white ball" and "the sample contains balls of both colors" independent?
1. E = contains at least white ball 2. F = both colors 3. Pr(E) = WW, RW, WR WW = 4/9 x 3/8 = 5/18 RW = 5/9 x 4/8 = 5/18 WR = 4/9 x 5/8 = 3/18 Total = 13/18 4. Pr(F) = RW, WR 5/18 + 5/18 = 10/18 10/18 does NOT equal 13/18, so they are NOT independent
Marbles question: Five red marbles and nine white marbles, sample of 4. What is the probability that, if at least one of the four is white, all four are white marbles?
1. Find the probability of all four marbles being white = 9/14 x 8/13 x 7/12 x 6/11 (have to decrease denominator, as white marbles are being taken without replacement) 2. Find the probability of all four being red marbles = 5/14 x 4/13 x 3/12 x 2/11 3. Take 1 - the answer to step 2 Answer is 0.1265
Roll a die and consider the following two events, E={1, 4, 5}, F={2, 5}. Are the events E and F independent?
1. Pr(E) = 3/6 = 1/2 2. Pr(F) = 2/6 =1/3 3. Pr(EnF) = 1/6 Pr(E) x Pr(F) = 1/6 Yes independent
Insurance: Retired man determines that the probability of living 5 more years is .9 He decides to take out a life insurance policy that will pay $10,000 in the event that he dies during the next 5 years. What is the expected value of the payout of this policy? How much should a person be willing to pay for this policy?
1. There is .9 chance he will live for the next 5 years, so he will receive $0 There is a .1 he will die in the next five years, so his estate will earn $10,000 2. 0.1 (10,000) + 0.9 (0) = $1000 Answer: a person should be willing to pay UP TO $1000 for this policy.
Carnival game: urn contains two red balls and four white balls Player picks one ball at a time out of the urn without replacement Game continues until a red ball is drawn Player pays $1 to pay and earns $0.50 for each ball drawn What is the probability distribution for the player's earning and find the expected value.
1. draw out tree diagram, with the first branch being 4/6 for drawing a white ball and 2/6 for drawing a red ball (the amount of $ won in this round if the person drew a red ball would be -$0.50) 2. continue the tree diagram until you have reaches 0 as the probability of drawing a white ball 3. E(X) = 1/3 (-0.50) + (2/3) (2/5) (0) + ... (The 2/3 > 2/5 is following the branches that always lead to red, as drawing a red ball ends the game and the money earned is calculated)
Percentile
70 percentile means that 70% of the population is below the measured value z = x-u/variance
Gambling odds question: Find total probability of four teams and then explain why it makes sense that the answer is over 1.
EX: if odds of winning is 12 to 1, then you have to calculate the odds of losing, so switch the order 1. Do this for all four teams and then find the least common denominator to calculate the total, which is always over 1. It is always over 1 in gambling so the bookie can make a profit regardless of whether the bettor wins or loses money
Suppose that we have a white urn containing six white balls and one red ball and we have a red urn containing one white ball and two red balls. An experiment consists of selecting at random a ball from the white urn and then (without replacing the first ball) selecting at random a ball from the urn having the color of the first ball. Find the probability that the second ball is red.
Tree diagram branches: White 1 = 6/7 -White 2 = 5/6 -Red 2 = 1/6 Red 1 = 1/7 -White 2 = 1/3 -Red 2 = 2/3 Second ball being red: White 1 >> Red 2 = (6/7) x (1/6) = 1/7 Red 1 >> Red 2 = (1/7) x (2/3) = 2/21 Answer: 1/7 + 2/21 = 5/21
