3.2 Independent and Mutually Exclusive Events

You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are 13 cards in each suit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J (jack), Q (queen), and K (king) of that suit. S = spades, H = Hearts, D = Diamonds, C = Clubs.

1. Suppose you pick four cards, but do not put any cards back into the deck. Your cards are QS, 1D, 1C,…

Flip two fair coins. Find the probabilities of the events.

1. Let F = the event of getting at most one tail (zero or one tail).
2. Let G = the event of getting two faces that are the same.
3. Let H = the event of getting a head on the first flip followed by a head or tail on the second flip.
4. Are F and G mutually exclusive?
5. Let J = the event of getting all tails. Are J and H mutually exclusive?

Are C and E mutually exclusive events? (Answer yes or no.) Why or why not?

Let event A = learning Spanish. Let event B = learning German. Then A AND B = learning Spanish and German. Suppose P(A) = 0.4 and P(B) = 0.2. P(A AND B) = 0.08. Are events A and B independent? Hint: You must show ONE of the following:

• P(A|B) = P(A)
• P(B|A) = P(B)
• P(A AND B) = P(A)P(B)

a. Show that P(G|H) = P(G).

b. Show P(G AND H) = P(G)P(H).

Let event C = taking an English class. Let event D = taking a speech class.

Suppose P(C) = 0.75, P(D) = 0.3, P(C|D) = 0.75 and P(C AND D) = 0.225.

Justify your answers to the following questions numerically.

1. Are C and D independent?
2. Are C and D mutually exclusive?
3. What is P(D|C)?

In a particular college class, 60% of the students are women. Fifty percent of all students in the class have long hair. Forty-five percent of the students are women and have long hair. Of the women students, 75% have long hair. Let W be the event that a student is a women. Let L be the event that a student has long hair. One student is picked randomly. Are the events of being a woman and having l…

1. Toss one fair coin (the coin has two sides, H and T). The outcomes are ________. Count the outcomes. There are ____ outcomes.
2. Toss one fair, six-sided die (the die has 1, 2, 3, 4, 5 or 6 dots on a side). The outcomes are ________________. Count the outcomes. There are ___ outcomes.
3. Multiply the two numbers of outcomes. The answer is _______.
4. If you flip one fair coin and follow it with the toss of…

J and K are independent events. P(J|K) = 0.3. Find P(J).

Q and R are independent events. P(Q) = 0.4 and P(Q AND R) = 0.1. Find P(R).

The probability that a person develops cancer is 0.4567. The probability that a person has at least one false positive test result (meaning the test comes back for cancer when the person does not have it) is 0.51. Some of the following questions do not have enough information for you to answer them. Write “not enough information” for those answers. Let C = a person develops cancer and P = a person…

Given events J and K: P(J) = 0.18; P(K) = 0.37; P(J OR K) = 0.45

1. Find P(J AND K).
2. Find the probability of the complement of event (J AND K).
3. Find the probability of the complement of event (J OR K).