PN1 PROBABILITY EXERCISE ANSWERS INTRODUCTION PN-11. A box contains six marbles: 3 red ®, 2 blue (B), and 1 white (W).One marble is drawn from the box. Use the scale below to describe the chance of each of the following events occurring. Not Slim Even Good Almost NO ÷||||||||||| ²CERTAINCHANCE Likely Chance Chance Chance SureA. A red marble is drawn. EVENB. A blue marble is drawn. SLIM or slightly better than slim.C. A white marble is drawn. NOT LIKELYD. A black marble is drawn. NO CHANCE !!!E. A blue or white marble is drawn. EVEN, because three of the six are winners.F. The marble drawn is not white. ALMOST SURE – only one way to lose.G. The marble drawn is not black. CERTAIN !!!2. If you were going to turn the above scale into a numeric scale, what values would you assign? Not Slim Even Good Almost NO ÷||||||||||| ²CERTAINCHANCE Likely Chance Chance Chance Sure3. What numbers would you assign to each of the following events?A. A red marble is drawn. HALF (half the marbles are winners)6B. A blue marble is drawn. '23 or '1 6C. A white marble is drawn. '1D. A black marble is drawn. 0/6 = 06E. A blue or white marble is drawn. '36F. The marble drawn is not white. '56G. The marble drawn is not black. '6 = 1 (Note this is 1 — P(black) ) 4. The possible outcomes in the above experiment could be called: R B WSome people would argue there are 3 possibilities, so the chance of each must be one-third.What is wrong with this analysis? How would you try to convince that person otherwise?These people are making the ASSUMPTION that the three outcomes are equally likely to occur.This assumption is not so.Ask them to consider the experiment of drawing a marble from a jar containing 100 marbles,with 98 red, 1 blue and 1 white. Would they rather bet on getting a RED marble or a WHITEmarble?5. A SAMPLE SPACE for an experiment is the set of all possible outcomes of the experiment.1List two different SAMPLE SPACES (SS) for the above experiment . 2A. B. 12312{ R, B, W } {R , R , R , B , B , W}Which SS would you rather use to analyze the chance of getting a White marble?The second SS has the advantage that all six outcomes are equally likely.Generally, the most detailed SS is the simplest!6. LIST SAMPLE SPACES for each of the following experiments: PN-2(use the back of page PN-1)A. Roll a fair die and see how many dots turn up. SS = { 1, 2, 3, 4, 5, 6 }B. Toss a fair coin and see what face turns up. SS = { H, T }C. Draw a card from an ordinary deck of cards. SS is bottom of this page.D. Roll a pair of fair dice.... See SS below right **E. Toss a fair coin twice and see what turns up each time. { HH, HT, TH, TT }F. Toss a fair coin three times.... {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }#6d** SS = { 11 12 13 14 15 16 7. In experiment 6A, what is the probability that... 21 22 23 24 25 26 6a. the die turns up 6 (dots)? '1 31 32 33 34 35 36 6b. the die turns up an even number of dots? '3 41 42 43 44 45 46 c. the die turns up 7 dots? 0 51 52 53 54 55 56 61 62 63 64 65 66 }8. In experiment 6C, what is the probability that the card drawn is ...52a. a “face card” (J or Q or K) ? '12 or 13'352b. a “heart”? '13 or 4'1c. You will win the big prize if the card is a face card or a heart. What is the probability you will win? 52'22 or 26'11d. What is the probability the card drawn is NOT a Face card or a Heart? 52'40 or 13'109. In experiment 6D, what is the probability that the sum of the two numbers on the dice ...36a. is 2? '1 36b. is 11? '2 36c. is 7? '6 or 6'136d. is 7 or 11? '8 or 9'2(In the game of craps, a sum of 7 or 11 on the first roll is an automatic win.)36e. is even? '18 or 2'110. In experiment 6E, what is the probability that...4a. two Heads turn up? '14b. one Head and one Tail turn up? '2 or 2'14c. the coin does not turn up Heads twice? '3SS for #6C: AÌ AË AÊ AÍ 2Ì 2Ë 2Ê 2Í 3Ì 3Ë 3Ê 3Í 4Ì 4Ë 4Ê 4Í 5Ì 5Ë 5Ê 5Í 6Ì 6Ë 6Ê 6Í 7Ì 7Ë 7Ê 7Í 8Ì 8Ë 8Ê 8Í 9Ì 9Ë 9Ê 9Í10Ì 10Ë 10Ê 10Í J Ì J Ë J Ê J ÍQÌ QË QÊ QÍKÌ K Ë KÊ KÍPROBABILITY OBSERVATIONS PN-3These are the “ground rules” of probability:A probability experiment is a repeatable action with a known set of possible outcomes, in which theparticular outcome is unknown in advance of each repetition of the experiment.The set of possible outcomes is the Sample Space.P1. Probabilities for the outcomes of the entire Sample Space add up to 1.P2. Probability is never negative. A zero probability means “can’t happen”, or “virtually impossible”.P3. Probability never exceeds 1. A probability of 1 means “certain” or “virtually certain”.In addition, we intuitively understand the following aspects of probability:P4. Probabilities ADD:Defn: If A1B = ö then A and B are disjoint. Defn: Events A & B for which P(A1B) = 0 are mutually exclusive..* IF A and B are mutually exclusive events for a probability experiment, then P(A or B) = P(A) + P(B) (This is one of two so-called "Addition Rules", which follow from the basic rules of probability.) "The more ways to win, the greater the chance of winning."P4A. Addition rule for M.E. events: P(A c B) = P(A) + P(B).See how this applies to #9d: P(7 or 11)=36 36P(7) + P(11) = ' + '62But some events are not so simply viewed... P4A. The general Addition Rule is: P( A or B) = P(A c B) = P(A) + P(B) ! P(A 1 B)See how this applies to #8c: P( Ì or F) = P(Ì) +P(F) — P(ÌF) 52 52 — 52 52= ' + '' = '13 12 3 22Addition rule for M.E. events is just a special case of the general Addition rule, where P(A 1 B) is 0 !P5. Defn: If two events A & B are disjoint, but together make up the entire sample space, then theevents are complementary. For instance, in experiment 6A, the events A, that the die turnsup 1, and B, that the die turns up more than 1, are complementary. B is the complement of A.6SS = { 1, 2, 3, 4, 5, 6} P (SS) = '6 or 16A = { 1 } P(A) = '16B = { 2, 3, 4, 5, 6 } P(B) = '5In experiment 6D, rolling a pair of fair dice, let’s call A the event that the sum of the dice is 12. In#9a, we found that P(A) = 1/36. What is the probability that the sum of …