Your pocket contains 2 quarters and 2 Nickels. You randomly pick one of the coins and without replacing it you pick a second coin.

Are the events, Q1 = First pick was a quarter
Q2 = Second pick was a quarter
a) Independent?
b) What is the probability of getting a quarter both times, P (Q1 and Q2)?
c) What is the probability of getting a quarter and then a nickel, P (Q and N)?

a) yes

b) If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.

2/4 * (2-1)/(4-1) = ?

c) 2/4 * 2/(4-1) = ?

To determine whether the events Q1 = First pick was a quarter and Q2 = Second pick was a quarter are independent, we need to consider whether the outcome of the first pick affects the outcome of the second pick.

a) Events Q1 and Q2 are not independent because the outcome of the first pick affects the outcome of the second pick. If the first pick is a quarter, there is one less quarter remaining in the pocket for the second pick.

b) To calculate the probability of getting a quarter both times, P (Q1 and Q2), we can use the multiplication rule for independent events. Since the events Q1 and Q2 are not independent in this case, we need to calculate the probability of each event separately and multiply them together.

P (Q1) = Number of ways to get a quarter on the first pick / Total number of possible outcomes on the first pick
P (Q1) = 2 quarters / 4 coins
P (Q1) = 1/2

Now, for the second pick, assuming the first pick was a quarter, there is only 1 quarter remaining along with 2 nickels.

P (Q2 | Q1) = Number of ways to get a quarter on the second pick, given that the first pick was a quarter / Total number of possible outcomes on the second pick, given that the first pick was a quarter
P (Q2 | Q1) = 1 quarter / 3 coins
P (Q2 | Q1) = 1/3

Therefore, P (Q1 and Q2) = P (Q1) * P (Q2 | Q1)
P (Q1 and Q2) = 1/2 * 1/3
P (Q1 and Q2) = 1/6

c) To calculate the probability of getting a quarter and then a nickel, P (Q and N), we need to consider the number of ways to get a quarter on the first pick and then a nickel on the second pick.

P (Q1 and N2) = Number of ways to get a quarter on the first pick * Number of ways to get a nickel on the second pick / Total number of possible outcomes on the second pick
P (Q1 and N2) = 2 quarters * 2 nickels / 4 coins
P (Q1 and N2) = 4/4
P (Q1 and N2) = 1

Therefore, the probability of getting a quarter and then a nickel is 1.

a) To determine if the events Q1 (First pick was a quarter) and Q2 (Second pick was a quarter) are independent, we need to check if the probability of Q2 happening is affected by whether or not Q1 has already happened.

Since there are only 4 coins in total and we do not replace the first coin, the probability of Q2 happening depends on whether or not Q1 has already occurred. If Q1 happened and a quarter was picked, then only 3 coins remain, and 1 of them is a quarter. If Q1 did not happen and a nickel was picked, then 3 coins also remain, but there are still 2 quarters. Therefore, the events Q1 and Q2 are dependent.

b) To calculate the probability of getting a quarter both times (P(Q1 and Q2)), we multiply the probability of Q1 happening (2 quarters out of 4 coins) by the probability of Q2 happening given that Q1 has already happened (1 quarter out of 3 coins remaining).

P(Q1 and Q2) = P(Q1) * P(Q2|Q1)
= (2/4) * (1/3)
= 2/12
= 1/6

So, the probability of getting a quarter both times is 1/6.

c) To calculate the probability of getting a quarter and then a nickel (P(Q and N)), we need to consider all possible combinations of events. There are two cases:

Case 1: First pick is a quarter, second pick is a nickel
In this case, the probability of picking a quarter first is 2/4, and the probability of picking a nickel second is 2/3 (since we do not replace the first coin).

P(Q and N) = P(Q1) * P(N2|Q1)
= (2/4) * (2/3)
= 4/12
= 1/3

Case 2: First pick is a nickel, second pick is a quarter
In this case, the probability of picking a nickel first is 2/4, and the probability of picking a quarter second is 2/3.

P(Q and N) = P(N1) * P(Q2|N1)
= (2/4) * (2/3)
= 4/12
= 1/3

Therefore, the total probability of getting a quarter and then a nickel is the sum of the two cases:

P(Q and N) = P(Q and N in Case 1) + P(Q and N in Case 2)
= 1/3 + 1/3
= 2/3

So, the probability of getting a quarter and then a nickel is 2/3.