Module 2 Activities-F2F Probability PDF

Module 2 – Probability Some common examples you may encounter when working with probabilities include (but are not limited to): playing cards, dice, coins, drawing marbles from a bag/urn, etc. If you are unfamiliar with any of these, there are many virtual generators available online. Here are some...

Module 2 – Probability Some common examples you may encounter when working with probabilities include (but are not limited to): playing cards, dice, coins, drawing marbles from a bag/urn, etc. If you are unfamiliar with any of these, there are many virtual generators available online. Here are some links to a few: Cards: https://www.random.org/playing-cards/ https://deck.of.cards Dice: https://www.random.org/dice/ Coins: https://randomwordgenerator.com/coin-flip.php Urn: https://planetcalc.com/7679/ Part 1: Terminology and the Basics of Probability 1. List the sample space of each of the following scenarios using S to indicate the sample space. For example, the sample space of rolling a single die is S={1, 2, 3, 4, 5, 6}. a. List the sample space of tossing a single coin. S={H,T} b. List the sample space of tossing two coins. S={HH,HT,TH,TT} c. List the sample space of tossing three coins (hint: there should be 8 elements in this list). S={HHH,HHT,HTH,THH,TTT,TTH,THT,HTT} d. List the sample space of drawing one marble from a bag containing 3 red, 2 green, and 5 yellow marbles and recording the color. S={R,G,Y} 2. Find the probabilities in each of the scenarios below: a. Probability that you will get heads on both coins when flipping 2 coins. ¼ or 0.25 b. Probability that you will get exactly 2 heads when flipping 3 coins. 3/8 or 0.375 1  c. Probability that you will get at least 2 heads when flipping 3 coins. ½ or 0.5 d. Probability of not getting a tail when you flip 3 coins. 1/8 or 0.125 e. If 5% of all computers sold in a given year will malfunction, what is the probability that a computer purchased that year will not malfunction? 0.95 or 95% f. What is the probability of rolling a 5 on a single dice roll? 1/6 or 0.16667 g. What is the probability of rolling a 5 or less on a single dice roll? 5/6 or 0.8333 h. What is the lowest and highest probability possible of an event? Explain. The lowest probability of an event is 0, which means the event is impossible and will never occur. The highest probability of an event is 1, which means the event is certain and will always occur. Part 2: Compound Events, Conditional Probability, & Independence Recall some of the key terms discussed in the lectures: Mutually exclusive events don’t overlap, i.e., P(A and B) = 0 Events are independent if the probability of one does not affect the probability of the other i.e., P(A|B) = P(A) and P(B|A) = P(B) Multiplication Rule: o P(A and B) = P(A)P(B) if A and B are independent. o P(A and B) = P(A) P(B|A) if A and B are not independent. Addition Rule: o P(A or B) = P(A) + P(B) – P(A and B) 2  1. Consider the following two events related to a deck of playing cards: 𝐴 = A card is an Ace 𝐵 = A card is red Describe in words what each of the following means and then state its value: a. P(𝐴|𝐵) 0.07692308 b. P(𝐵|𝐴) 0.5 c. P(𝐴 and 𝐵) 0.03846154 d. P(𝐴 or 𝐵! ) 0.53846154 2. Consider a bag that containing 3 pink gumballs, 5 orange gumballs, and 2 blue gumballs. a. Answer the following if you draw one gumball. i. What is the sample space? S={P,O,B} ii. Are the outcomes equally likely? Explain. No, because there are different amounts of each respective color. iii. What are the probabilities of drawing each color? P(Pink)=3/10 P(Orange)=1/2 P(Blue)=1/5 iv. What is the probability you draw a pink or an orange gumball? 4/5 or 0.8 v. What rule did you use in the above question? Addition rule 3  3. Considering the same distribution of gumballs as in question 2 above (3 pink gumballs, 5 orange gumballs, and 2 blue gumballs) answer the following if you draw two gumballs without replacement, recording their colors in order. a. How many elements will be in the sample space? (Hint: draw a tree diagram) 9 b. Are the outcomes equally likely? Explain. No, the outcomes are not equally likely because the number of gumballs of each color is different, which affects the likelihood of drawing each color. c. What is the probability of drawing a pink gumball on your second draw given that you first drew an orange gumball? 1/3 or 0.3333 d. What is the probability that on your first draw you pick a pink gumball and then pick an orange gumball on your second draw? 1/6 or 0.1667 Part 3: Additional Examples 1. Calculate the probabilities below: a. P(Drawing 3 cards from a standard deck of 52 cards (without replacement) and all 3 of them are aces) P(3 aces)= 4/52 * 3/51 * 2/50 = 1/5525 or 0.000181 b. P(Drawing first a spade and then a heart (without replacement) when drawing 2 cards from a standard deck of 52 cards) P(1st is spade)=13/52 * P(2nd is heart)=13/51 = 13/204 or 0.0637 c. P(Drawing 6 cards (without replacement), and none of them are a queen) P(1st is not queen)=48/52 Etc. P(none are queens)=0.602 d. P(In a litter of 4 puppies, all 4 of them are male) 1/16 or 0.0625 4  e. P(Rolling a 6, three times in a row on a 6-sided die) 1/6*1/6*1/6=1/216 or 0.00463 2. Which of the repeated events in the question above are independent? Which are dependent? Explain. Dependent Events: o Drawing 3 aces from a deck without replacement o Drawing a spade first, then a heart (without replacement) o Drawing 6 cards and none being queens (without replacement) Independent Events: o In a litter of 4 puppies, all 4 being male o Rolling a 6 three times in a row on a 6-sided die 3. How many cards must you draw so that the probability of drawing “at least 1 red card” is 1? 27 4. Consider two events A and B in the same sample space where P(A) = 0.37 and P(B) = 0.89 a. Why is it impossible for events A and B to be mutually exclusive? It is impossible for events A and B to be mutually exclusive because the sum of their probabilities exceeds 1, which is not possible in probability theory. b. Assuming A and B are independent, calculate P(A and B). P(A∩B)=P(A)×P(B)=0.37×0.89=0.3293 c. Using the value from b) above, calculate P(A or B). P(A∪B)=0.37+0.89−0.3293=0.9307 5. If you were to roll 2 dice and consider their sum an event, what are their associated probabilities? Is each sum equally likely? Sum 2 3 4 5 6 7 8 9 10 11 12 P(Sum) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 *Note: The probabilities should sum to 1. 5 Part 4: Contingency tables 1. Consider the following scenario regarding a group surveyed on their favorite sandwich shop. Using the table below calculate the associated probabilities: Half Fast Snarf’s Potbelly Total Under 25 240 180 45 465 Over 25 23 159 110 292 TOTAL 263 339 155 757 a. P(Over 25) 0.3857 b. P(Half Fast Subs or Under 25) 0.74827982 c. P(Snarf’s and Potbelly) 0.09169373 d. P(Under 25|Half Fast) 0.61426684 e. P(Snarf’s|Over 25) 0.54452055 f. P(Over 25|Potbelly) 0.70967742 g. If 3 individuals are selected, what is P(all 3 favored Potbelly) 6  2. Consider the following contingency table related to college major: Math Business Chemistry Total Female 22 67 58 147 Male 49 73 20 142 TOTAL 71 140 78 289 Let 𝐵 = the student is majoring in Business and 𝑀 = a student is male, find the following: a. P(𝑀|𝐵) 0.52142857 b. P(𝑀! and 𝐵) 0.23183391 c. P(𝑀 or 𝐵! ) 0.7535949 d. Are events 𝐵 and 𝑀 independent? No 3. Consider the following contingency table on aggression related to ice cream consumption and answer the associated questions. Let M = Mild aggression, C = Consumed Ice Cream, and D = Did Not Consume Ice Cream Consumed Ice Did not Consume Ice Total Cream Cream Mild Aggression 60 90 150 Moderate Aggression 30 40 70 Severe Aggression 10 20 30 Total 100 150 250 a. P(M) 0.6 7  b. P(C) 0.4 c. P(D) 0.6 d. P(M|C) 0.6 e. P(M|D) 0.6 f. Are events M and C independent? Yes 8