Question
Find the following probability.
James has 3 dimes, 4 pennies, and 2 quarters in his pocket. If each coin is equally likely to be pulled out of his pocket in order without replacement, what is the probability that he will pull out the 2 quarters in a row first?
Question 5 options:
0.525
0.0278
0.0634
0.256
James has 3 dimes, 4 pennies, and 2 quarters in his pocket. If each coin is equally likely to be pulled out of his pocket in order without replacement, what is the probability that he will pull out the 2 quarters in a row first?
Question 5 options:
0.525
0.0278
0.0634
0.256
Answers
GPT-4o mini
To find the probability that James will pull out the 2 quarters in a row first, we first determine the total number of coins James has:
- 3 dimes
- 4 pennies
- 2 quarters
Total coins = \(3 + 4 + 2 = 9\) coins.
We need to find the probability that the first two coins drawn are the two quarters.
1. **Calculating the probability of the first draw:** The probability that the first coin drawn is a quarter is:
\[
P(\text{first is quarter}) = \frac{2}{9}
\]
2. **Calculating the probability of the second draw:** After drawing the first quarter, there is now 1 quarter left and a total of 8 coins remaining. The probability that the second coin drawn is also a quarter is:
\[
P(\text{second is quarter | first is quarter}) = \frac{1}{8}
\]
3. **Calculating the joint probability:** The overall probability of drawing the two quarters in succession is the product of the two probabilities:
\[
P(\text{two quarters in a row}) = P(\text{first is quarter}) \times P(\text{second is quarter | first is quarter}) = \frac{2}{9} \times \frac{1}{8} = \frac{2}{72} = \frac{1}{36}
\]
Calculating \( \frac{1}{36} \) gives us approximately \( 0.0278 \).
Therefore, the probability that James will pull out the 2 quarters in a row first is \( \boxed{0.0278} \).
- 3 dimes
- 4 pennies
- 2 quarters
Total coins = \(3 + 4 + 2 = 9\) coins.
We need to find the probability that the first two coins drawn are the two quarters.
1. **Calculating the probability of the first draw:** The probability that the first coin drawn is a quarter is:
\[
P(\text{first is quarter}) = \frac{2}{9}
\]
2. **Calculating the probability of the second draw:** After drawing the first quarter, there is now 1 quarter left and a total of 8 coins remaining. The probability that the second coin drawn is also a quarter is:
\[
P(\text{second is quarter | first is quarter}) = \frac{1}{8}
\]
3. **Calculating the joint probability:** The overall probability of drawing the two quarters in succession is the product of the two probabilities:
\[
P(\text{two quarters in a row}) = P(\text{first is quarter}) \times P(\text{second is quarter | first is quarter}) = \frac{2}{9} \times \frac{1}{8} = \frac{2}{72} = \frac{1}{36}
\]
Calculating \( \frac{1}{36} \) gives us approximately \( 0.0278 \).
Therefore, the probability that James will pull out the 2 quarters in a row first is \( \boxed{0.0278} \).