4.6 Exercises

Suppose that you have 8 cards. 5 are green and 3 are yellow. The 5 green cards are numbered 1, 2, 3, 4, and 5. The 3 yellow cards are numbered 1, 2, and 3. The cards are well shuffled. You randomly draw one card.

• $G$ = card drawn is green
• $E$ = card drawn is even-numbered

• a List the sample space.
• b $\mathrm{P(G)\; =}$
• c $\mathrm{P(G|E)\; =}$
• d $\mathrm{P(G AND E)\; =}$
• e $\mathrm{P(G OR E)\; =}$
• f Are $G$ and $E$ mutually exclusive? Justify your answer numerically.

Refer to the previous problem. Suppose that this time you randomly draw two cards, one at a time, and with replacement .

• $G_1\text{= first card is green}$
• $G_2\text{= second card is green}$

• a Draw a tree diagram of the situation.
• b $P(G_1\text{ AND }{G}_2)=$
• c $P(\text{at least one green})=$
• d $P(G_2\mid G_1)=$
• e Are $G_2$ and $G_1$ independent events? Explain why or why not.

Refer to the previous problems. Suppose that this time you randomly draw two cards, one at a time, and without replacement .

• $G_1$ = first card is green
• $G_2$ = second card is green

• a Draw a tree diagram of the situation.
• b> $\mathrm{P(}{G}_1\text{ AND }{G}_2\mathrm{)\; =}$
• c $\text{P(at least one green) =}$
• d $\mathrm{P(}{G}_2|G_1\mathrm{)\; =}$
• e Are $G_2$ and $G_1$ independent events? Explain why or why not.

Roll two fair dice. Each die has 6 faces.

• a List the sample space.
• b Let $A$ be the event that either a 3 or 4 is rolled first, followed by an even number. Find $\mathrm{P(A)}$ .
• c Let $B$ be the event that the sum of the two rolls is at most 7. Find $\mathrm{P(B)}$ .
• d In words, explain what “ $\mathrm{P(A|B)}$ ” represents. Find $\mathrm{P(A|B)}$ .
• e Are $A$ and $B$ mutually exclusive events? Explain your answer in 1 - 3 complete sentences, including numerical justification.
• f Are $A$ and $B$ independent events? Explain your answer in 1 - 3 complete sentences, including numerical justification.

A special deck of cards has 10 cards. Four are green, three are blue, and three are red. When a card is picked, the color of it is recorded. An experiment consists of first picking a card and then tossing a coin.

• a List the sample space.
• b Let $A$ be the event that a blue card is picked first, followed by landing a head on the coin toss. Find $\text{P(A)}$ .
• c Let $B$ be the event that a red or green is picked, followed by landing a head on the coin toss. Are the events $A$ and $B$ mutually exclusive? Explain your answer in 1 - 3 complete sentences, including numerical justification.
• d Let $C$ be the event that a red or blue is picked, followed by landing a head on the coin toss. Are the events $A$ and $C$ mutually exclusive? Explain your answer in 1 - 3 complete sentences, including numerical justification.

An experiment consists of first rolling a die and then tossing a coin:

• a List the sample space.
• b Let $A$ be the event that either a 3 or 4 is rolled first, followed by landing a head on the coin toss. Find $\text{P(A)}$ .
• c Let $B$ be the event that a number less than 2 is rolled, followed by landing a head on the coin toss. Are the events $A$ and $B$ mutually exclusive? Explain your answer in 1 - 3 complete sentences, including numerical justification.

An experiment consists of tossing a nickel, a dime and a quarter. Of interest is the side the coin lands on.

• a List the sample space.
• b Let $A$ be the event that there are at least two tails. Find $\text{P(A)}$ .
• c Let $B$ be the event that the first and second tosses land on heads. Are the events $A$ and $B$ mutually exclusive? Explain your answer in 1 - 3 complete sentences, including justification.