I have 3 nickels, 1 quarter, and 1 penny in my pocket. If I pick out a coin from my pocket, do not replace it, then pick out a second coin from my pocket, what is the probability that the second coin that I pick out of my pocket is a nickel? So far I have come up with this: P(1st coin is a nickel)= 3/5. P(2nd coin is a nickel, if the first coin was a nickel)= 2/4. P(1st coin is a quarter)= P(1st coin is a penny)= 1/5. P(2nd coin is a nickel, if first coin was a quarter)= P(2nd coin is a nickel, if first coin was a penny)= 3/4. I don't know where to go from here.
To find the probability that the second coin you pick out of your pocket is a nickel, you need to consider all the possible scenarios and calculate the probabilities accordingly.
1. If the first coin you picked out is a nickel:
- The probability of picking a nickel as the first coin is 3/5.
- After picking a nickel, there are now 2 nickels left out of 4 remaining coins.
- Therefore, the probability of picking a nickel as the second coin, given that the first coin was a nickel, is 2/4.
2. If the first coin you picked out is not a nickel:
- There are two possible cases for the first coin being either a quarter or a penny.
2.1. If the first coin is a quarter:
- The probability of picking a quarter as the first coin is 1/5.
- After picking a quarter, there are no nickels left in the remaining 4 coins.
- Therefore, the probability of picking a nickel as the second coin, given that the first coin was a quarter, is 0.
2.2. If the first coin is a penny:
- The probability of picking a penny as the first coin is 1/5.
- After picking a penny, there are 3 nickels left out of 4 remaining coins.
- Therefore, the probability of picking a nickel as the second coin, given that the first coin was a penny, is 3/4.
Now, you need to calculate the overall probability by considering both the cases where the first coin is a nickel and where it is not a nickel:
Overall probability = (Picking a nickel as the first coin) * (Probability of picking a nickel as the second coin, given that the first coin was a nickel)
+ (Picking a quarter as the first coin) * (Probability of picking a nickel as the second coin, given that the first coin was a quarter)
+ (Picking a penny as the first coin) * (Probability of picking a nickel as the second coin, given that the first coin was a penny)
= (3/5) * (2/4) + (1/5) * (0) + (1/5) * (3/4)
Simplifying this expression, the probability that the second coin you pick out of your pocket is a nickel is:
= 6/20 + 0 + 3/20
= 9/20
Therefore, the probability that the second coin you pick out is a nickel is 9/20.
To find the probability of picking a nickel as the second coin, you need to consider the two scenarios:
1. The first coin you pick is a nickel:
In this scenario, you correctly determined that the probability of picking a nickel as the first coin is 3/5. After removing the nickel from your pocket, there are now 4 coins left (2 nickels, 1 quarter, and 1 penny). So, the probability of picking a nickel as the second coin, given that the first coin was a nickel, is 2/4.
2. The first coin you pick is not a nickel:
There are two possibilities here: either the first coin is a quarter or a penny. The probability of picking a quarter as the first coin is 1/5, and the probability of picking a penny as the first coin is also 1/5. After removing the non-nickel from your pocket, you will have 3 coins left (2 nickels and 1 quarter if you picked a quarter, or 2 nickels and 1 penny if you picked a penny). In both cases, the probability of picking a nickel as the second coin, given that the first coin was not a nickel, is 2/3.
Now, you can calculate the total probability of picking a nickel as the second coin by considering both scenarios:
P(Second coin is a nickel) = P(1st coin is a nickel) * P(2nd coin is a nickel, given that the first coin was a nickel)
+ P(1st coin is not a nickel) * P(2nd coin is a nickel, given that the first coin was not a nickel)
= (3/5) * (2/4) + [(1/5) + (1/5)] * (2/3)
Simplifying this equation will give you the probability that the second coin you pick is a nickel.