Question

Suppose we have a special deck of 13 cards: 2,3,4,5,6,7,8,9,10, jack, queen, king, ace. A face card is a card that has a face on it - jack (J), queen (Q) or king (K). An ace is worth 11 points, 2 - 10 are worth their point values (e.g., the 4 is worth 4 points), and jack, queen, and king are each worth 10 points. Suppose we draw a single card at random from this deck. Let A= "the card we draw has value greater than 8 ". Let B= "the card we draw has an even value". Let C= "the card we draw is a face card". a) [1 mark] Draw a Venn diagram. Ensure "ace" is labelled "ace" instead of "A", to avoid confusion. Put the experimental outcomes in the proper locations. Label events A, B, and C. Hint: Are you wondering what is meant by, say, "label event A"? It means "draw a cirele inside the rectangle, put the experimental outcomes associated with A inside the circle, and put the letter " A " in the circle". There are many examples of Venn diagrams in the Module 4 Review questions and Module 4 Tutorial questions. You may either draw your Venn diagram in Word or by hand. Please see the first page of this document for instructions on how to insert a picture of your handwritten work into Word if necessary. Yes, it's perfectly fine for some regions of the Venn diagram to be empty; yes, it's fine to list out the experimental outcomes that belong to each event, but it's not required for this question. b) [3 marks] Find P(C∣A). Show your work: show the equation you use, how you plug in, and your final answer rounded to 4 decimal places. You may check your work with Excel, but do not provide any Excel commands here. Note: when rounding, you must use the "approximately equal to" (≈) sign. You may draw an additional Venn diagram to assist you, but you must use an appropriate equation to answer this question. c) [3 marks] Calculate P(A∣( B ˉ ∩ C ˉ )). Show your work: show the equation you use, how you plug in, and your final answer. Provide your answer as a decimal, not a fraction. You may eheck your work with Excel, but do not provide any Excel commands here. Hint: P(A∩( B ˉ ∩ C ˉ ))=P(A∩ B ˉ ∩ C ˉ ). The same probability rules that applied to a set of two events also apply to sets of three events as we saw in the Module 4 Review questions. d) [3 marks] Are any two pairs of events (A,B),(A,C), or (B, C) mutually excluaive? Why or why not? Use the mathematical definition of mutual exclusivity. Show your work. Round probabilities to 4 decimal places.

Answer 1

The question deals with probabilities involving drawing cards and applying Venn diagrams and conditional probabilities. The calculations include P(C|A) and P(A|B ∩ C)) values and determine that no two events (A, B, C) are mutually exclusive.

Step-by-step explanation:

The student's question is addressing the concept of probability using Venn diagrams and conditional probabilities. Let A be 'the card we draw has value greater than 8', B be 'the card we draw has an even value', and C be 'the card we draw is a face card'.

Cards with a value greater than 8 (A) include: 9, 10, J, Q, K, and ace. Cards with an even value (B) are: 2, 4, 6, 8, and 10. Face cards (C) are: J, Q, K.

To solve part b, P(C|A) is the probability of C given A. We see that there are 6 cards in A and 3 of them are face cards. So, P(C|A) ≈ 0.5000 (3/6).

For part c, P(A|(B ∩ C)) is the probability of A given the card is neither even (B) nor a face card (C). Only the ace fits these criteria, so when the only outcome is the ace, P(A|(B ∩ C)) = 1.0000.

For part d, by evaluating the overlap of the sets, no two pairs from A, B, and C are mutually exclusive because all pairs have elements in common. For example, the 10 is both greater than 8 (A) and an even card (B).