A drawer contains 7 white shirts and 3 red shirts. What is the probability of picking a white shirt and then a red shirt, if the first shirt is not put back in?

Answer

To solve the problem of finding the probability of picking a white shirt followed by a red shirt without replacement, we can break it down into clear steps:

### Step 1: Determine the Total Number of Shirts
The drawer contains:
- 7 white shirts
- 3 red shirts

**Total number of shirts = 7 (white) + 3 (red) = 10 shirts.**

### Step 2: Calculate the Probability of Picking a White Shirt
The probability of picking a white shirt first is given by the formula:

\[ P(\text{White}) = \frac{\text{Number of white shirts}}{\text{Total number of shirts}} = \frac{7}{10} \]

### Step 3: Update the Total Number of Shirts After Picking a White Shirt
Since we are not replacing the first shirt, after picking one white shirt, the total number of shirts left is:

**Total shirts left = 10 - 1 = 9 shirts.**

### Step 4: Calculate the Probability of Picking a Red Shirt After Picking a White Shirt
Now, we need to find the probability of picking a red shirt after having already picked a white shirt. The number of red shirts remains the same (3), but the total number of shirts is now 9.

The probability of picking a red shirt now is:

\[ P(\text{Red | White}) = \frac{\text{Number of red shirts}}{\text{Total number of shirts left}} = \frac{3}{9} = \frac{1}{3} \]

### Step 5: Calculate the Combined Probability
The combined probability of both events (picking a white shirt first and then a red shirt) is the product of the individual probabilities:

\[ P(\text{White and then Red}) = P(\text{White}) \times P(\text{Red | White}) = \frac{7}{10} \times \frac{1}{3} \]

Calculating this gives:

\[ P(\text{White and then Red}) = \frac{7 \times 1}{10 \times 3} = \frac{7}{30} \]

### Final Answer
Thus, the probability of picking a white shirt followed by a red shirt without replacement is:

\[ \frac{7}{30} \]

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