Michael has 3 quarters, 2 dimes, and 3 nickels in his pocket. He randomly draws two coins from his pocket, one at a time, and they are both dimes. He says the probability of that occurring is 1

Michael's reasoning is incorrect. To determine the probability of drawing two dimes in succession, we need to consider the total number of coins and the correct way to calculate the probability of drawing dimes.

Let's analyze the situation:

1. Count the Total Coins:

   • Quarters: 3
   • Dimes: 2
   • Nickels: 3
   • Total coins = 3 (quarters) + 2 (dimes) + 3 (nickels) = 8 coins.

2. Calculate the Probability of Drawing Two Dimes:

   • The probability of drawing a dime on the first draw is the number of dimes divided by the total number of coins: \[ P(\text{first dime}) = \frac{2}{8} = \frac{1}{4} \]

   • After drawing one dime, there is now 1 dime left and a total of 7 coins remaining. So, the probability of drawing a second dime after having drawn the first is: \[ P(\text{second dime | first dime}) = \frac{1}{7} \]

3. Combine the Probabilities:

   • The total probability of both events (drawing two dimes in succession) is the product of the probabilities of each event: \[ P(\text{two dimes}) = P(\text{first dime}) \times P(\text{second dime | first dime}) = \frac{1}{4} \times \frac{1}{7} = \frac{1}{28} \]

Thus, the correct probability of randomly drawing two dimes from Michael's pocket is \(\frac{1}{28}\), not \(\frac{1}{4}\) as Michael suggested.