Probability of Drawing a Dime and a Nickel After Replacing the First Draw
This article discusses the probability of drawing a specific combination of coins (a dime and a nickel) from a mixed set of coins, with the condition that the first coin is replaced before the second draw. We will use a detailed step-by-step method to determine the probability.
Identifying the Total Number of Coins
Rickey has 3 quarters, 5 dimes, and 2 nickels in his pocket. By adding these, we can find the total number of coins:
Total coins 3 quarters 5 dimes 2 nickels[/itex]Total coins 10
Calculating the Probabilities
To find the probability that Rickey will draw a dime followed by a nickel, we need to calculate the probability of each event separately and then multiply them, since these are independent events.
Probability of Drawing a Dime First
The probability of drawing a dime on the first draw is the ratio of the number of dimes to the total number of coins:
P(Dime) Number of DimesTotal Coins 510 12[/itex]The probability of drawing a dime first is: 1/2
Probability of Drawing a Nickel Second (After Replacement)
After replacing the first coin, the total number of coins remains the same. Therefore, the probability of drawing a nickel on the second draw is:
P(Nickel) Number of NickelsTotal Coins 210 15[/itex]The probability of drawing a nickel second is: 1/5
Combining the Probabilities
Since the draws are independent, the combined probability of both events happening is the product of their individual probabilities:
P(Dime and Nickel) P(Dime) times; P(Nickel) 12 times; 15 110[/itex]The combined probability of drawing a dime and then a nickel is: 1/10
Understanding Independent Events
When the first coin is replaced, the events are independent, which means the outcome of the second draw does not depend on the first. This is crucial for calculating probabilities in such scenarios.
Related Calculations
As Rickey is replacing the coin after the first draw, this is two independent events. The probability of drawing a nickel the first time is 2 in 10. After the replacement, the probability of drawing a nickel the second time is also 2 in 10. The combined probability is the product of the two probabilities: 2/10 times; 2/10 4 in 100 0.04 or 4%. This confirms our earlier calculation that the probability of drawing a dime first and then a nickel is 1/10.
Conclusion
In conclusion, the probability of Rickey drawing a dime and a nickel from his pocket, with the first coin being replaced before drawing the second, is 1/10. This is a clear example of calculating probabilities for independent events in a real-world scenario.