To find the probability of picking a dime exactly twice when randomly drawing from the bag three times (with replacement), we can use the binomial probability formula:
\[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \]
Where:
- \( n = 3 \) (the number of trials)
- \( k = 2 \) (the number of successes, which in this case is picking a dime)
- \( p \) is the probability of success on an individual trial.
First, we identify the probabilities:
- The total number of coins = 3 (a nickel, a penny, and a dime).
- The probability of picking a dime \( p = \frac{1}{3} \).
- The probability of not picking a dime \( 1-p = \frac{2}{3} \).
Now we substitute the values into the binomial formula:
\[ P(X=2) = \binom{3}{2} \left(\frac{1}{3}\right)^2 \left(\frac{2}{3}\right)^{3-2} \]
Calculating the binomial coefficient:
\[ \binom{3}{2} = 3 \]
Now substituting all the values:
\[ P(X=2) = 3 \left(\frac{1}{3}\right)^2 \left(\frac{2}{3}\right)^{1} \]
Calculating each term:
\[ = 3 \cdot \frac{1}{9} \cdot \frac{2}{3} \] \[ = 3 \cdot \frac{2}{27} \] \[ = \frac{6}{27} \] \[ = \frac{2}{9} \]
Thus, the probability that a dime is picked exactly twice is:
\[ \frac{2}{9} \]
The correct answer is:
Start Fraction 2 over 9 End Fraction.