Probability #11 - Questions and Answers

Question 1 Suppose the probability of winning the Powerball lottery is 0.1, while the probability of being abducted by aliens is 0.6. If these two events are independent, what is the probability of winning Powerball but not being abducted? A) 0.06 B) 0.54 C) 0.04 D) 0.6 E) 0 F) 0.1 G) 1 H) 0.36 Answer Step 1 The probability of winning the Powerball lottery is 0.1. That is, P(Winning) = 0.1, and the probability of being abducted by aliens is 0.6, then the probability of not being abducted by aliens is 0.4(= 1 - 0.6). That is, P(Not abducted) = 0.4. The two events are independent. Step 2 The probability of winning Powerball but not being abducted is, P(Winning ∩ Not abducted) = P(Winning)P(Not abducted) = 0.1 * 0.4 = 0.04 Thus, the probability of winning Powerball but not being abducted is 0.04. Correct Answer: Option (C) 0.04. Question 2 You randomly select one card from a standard deck. (a) What is the probability of selecting a king or a queen? (b) What is the probability of selecting a face card or a 10? (c) What is the probability of selecting a spade or a heart? (d) What is the probability of selecting a red card or a black card? Answer Step 1 Given that one card is drawn from a standard deck. Probability is defined as: P(x) = # of favorable outcomes / # of possible outcomes (a) No. of cards is 52 No. of kings from each suit is 4 No. of queens from each suit is 4 The no of favorable cases is 4 + 4 = 8 The probability of picking either of them is 8/52 (b) No. of cards is 52 No. of face cards from each suit is 4 x 3 = 12 card have no. 10 from each suit is 4 The no. of favorable cases is 12 + 4 = 16 The probability of picking either of them is 16/52 Step 2 (c) No. of cards is 52 No. of spade cards = 13 No. of heart cards = 13 The no. of favorable cases is 13 + 13 = 26 The probability of picking either of them is 26/52 (c) No. of cards is 52 No. of red cards of heart & diamond = 13 + 13 = 26 No. of black cards of spade & clubs = 13 + 13 = 26 The no of favorable cases is 26 + 26 = 52 The probability of picking either of them is 52/52 = 1 Question 3 Different hands are possible while playing 7-card poker with a conventional 52-card deck, 52C7, or 133,784,560. The likelihood of receiving different hands is calculated by dividing the total number of possible outcomes by 133,784,560. The probabilities and number of possible outcomes for each type of hand are displayed to the right. Calculate the likelihood of not receiving this kind of hand. Number of ways the hand can occur 7480 probability 7480/133,784,560 Answer Step 1 For any event A and the sample space S, the probability can be found as, P(A) = n(A) / n(S) = Number of Favorable cases corresponding to event A / Total number of cases Step 2 The Probability of complement of the event A is calculated as, P(Aᶜ) = 1 - P(A) Step 3 In a 7 card poker, a particular type of hand can occur in 7,480 ways. Let A be the event of the particular type of hand. The probability of A is, P(A) = n(A) / n(S) = 7,480 / 133,784,560 Step 4 The event of not being dealt of this type of hand is denoted by Aᶜ. The probability of Aᶜ can be found as, P(Aᶜ) = 1 - P(A) = 1 - 7,480 / 133,784,560 = 133,777,080 / 133,784,560 = 0.999944 Step 5 Answer: The probability of not being dealt this type of hand is 0.999944. Question 4 Here is the probability model for the blood type of a randomly chosen person in the United States. Blood type | O | A | B | AB | Probability | 0.53 | 0.18 | 0.01 | 0.28 | What is the probability that a randomly chosen American does not have type O blood? Answer Step 1 Given that the probability of type O blood is = 0.53 So the probability of not having type O blood is = 1 - 0.53 = 0.47 = 47% [using the complement property of probability] Step 2 Answer: 47% Question 5 A card is drawn randomly from a standard 52-card deck. Find the probability of the given event. a) The card drawn is 3 The probability is: ? b) The card drawn is a face card (Jack, Queen, or King) The probability is: ? c) The card drawn is not a face card. The probability is: ? Answer Step 1 The number of cards in a deck of cards is 52. Step 2 a) The number of 3 numbered cards is 4. The probability of the card drawn is 3 is 4/52 = 1/13 = 0.0769. b) The number of face cards is 3 * 4 = 12. The probability of the card drawn is a face card is 12/52 = 3/13 = 0.2308 b) The number of face cards is 3 * 4 = 12. The number of not face cards is 52 - 12 = 40. The probability of the card drawn is not a face card is 40/52 = 10/13 = 0.7692