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), K (king) of that suit. Three cards are picked at random.

1. Suppose you know that the picked cards are Q of spades, K of hearts and Q of spades. Can you decide if the sampling wa…

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,…

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. Suppose that you sample four cards without replacement. Which of the following outcomes are possibl…

Draw two cards from a standard 52-card deck with replacement. Find the probability of getting at least one black card.

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?

A box has two balls, one white and one red. We select one ball, put it back in the box, and select a second ball (sampling with replacement). Find the probability of the following events:

1. Let F = the event of getting the white ball twice.
2. Let G = the event of getting two balls of different colors.
3. Let H = the event of getting white on the first pick.
4. Are F and G mutually exclusive?
5. Are G and H mut…

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 $A \cap B =$ learning Spanish and German. Suppose $P \left(A\right) = 0.4$ and $P \left(B\right) = 0.2$. $P \left(A \cap B\right) = 0.08$. Are events A and B independent? Hint: You must show ONE of the following:

• $P \left(\right. A \left|\right. B \left.\right) = P \left(A\right)$
• $P \left(\right. B \left|\right. A \left.\right) = P \left(B\right)$
• $P \left(A \cap B\right) = P \left(A\right) P \left(B\right)$

a. Show that $P \left(\right. G \left|\right. H \left.\right) = P \left(G\right)$.

b. Show $P \left(G \cap H\right) = P \left(G\right) P \left(H\right)$.

In a bag, there are six red marbles and four green marbles. The red marbles are marked with the numbers 1, 2, 3, 4, 5, and 6. The green marbles are marked with the numbers 1, 2, 3, and 4.

• R = a red marble
• G = a green marble
• O = an odd-numbered marble
• The sample space is S = {R1, R2, R3, R4, R5, R6, G1, G2, G3, G4}.

S has ten outcomes. What is $P \left(G \cap O\right)$?

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

Suppose $P \left(C\right) = 0.75$, $P \left(D\right) = 0.3$, $P \left(\right. C \left|\right. D \left.\right) = 0.75$ and $P \left(C \cap D\right) = 0.225$.

Justify your answers to the following questions numerically.

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

A student goes to the library. Let events B = the student checks out a book and D = the student checks out a DVD. Suppose that $P \left(B\right) = 0.40$, $P \left(D\right) = 0.30$ and $P \left(B \cap D\right) = 0.20$.

1. Find $P \left(\right. B \left|\right. D \left.\right)$.
2. Find $P \left(\right. D \left|\right. B \left.\right)$.
3. Are B and D independent?
4. Are B and D mutually exclusive?

In a basketball arena,

• 70% of the fans are rooting for the home team.
• 25% of the fans are wearing blue.
• 20% of the fans are wearing blue and are rooting for the away team.
• Of the fans rooting for the away team, 67% are wearing blue.

Let A be the event that a fan is rooting for the away team.
Let B be the event that a fan is wearing blue.
Are the events of rooting for the away team and wearing bl…

Mark is deciding which route to take to work. His choices are I = the Interstate and F = Fifth Street.

• $P \left(I\right) = 0.44$ and $P \left(F\right) = 0.56$
• $P \left(I \cap F\right) = 0$ because Mark will take only one route to work.

What is the probability of $P \left(I \cup F\right)$?

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…

A box has two balls, one white and one red. We select one ball, put it back in the box, and select a second ball (sampling with replacement). Let T be the event of getting the white ball twice, F the event of picking the white ball first, S the event of picking the white ball in the second drawing.

1. Compute $P \left(T\right)$.
2. Compute $P \left(\right. T \left|\right. F \left.\right)$.
3. Are T and F independent?.
4. Are F and S mutually exclusive?
5. Are F and S independ…

$J$ and $K$ are independent events. $P \left(\right. J \left|\right. K \left.\right) = 0.3 .$ Find $P \left(J\right) .$

$Q$ and $R$ are independent events. $P \left(\text{Q}\right) = 0.4$ and $P \left(Q \cap R\right) = 0.1 .$ Find $P \left(R\right) .$

Find the probability that the selected occupation has an Emotional Health Index Score of 81.0.

Find the probability that the selected occupation has an Emotional Health Index Score between 80.5 and 82.

What is the probability that an occupation has an Emotional Health Index Score of 80.7 or 82.7?

What occupation has the highest emotional index score?

What is the range of the data?

If all occupations are equally likely for a certain individual, what is the probability that they will have an occupation with lower than average EHIS?

The probability that someone develops cancer during their lifetime 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 ca…

Given events $J$ and $K : P \left(J\right) = 0.18 ;$ $P \left(K\right) = 0.37 ;$ $P \left(J \cup K\right) = 0.45$

1. Find $P \left(J \cap K\right) .$
2. Find the probability of the complement of event $\left(J \cap K\right) .$
3. Find the probability of the complement of event $\left(J \cap K\right) .$