Questions: Two six-sided die are rolled. The figure shows the 36 equally likely outcomes. Find the theoretical probability of rolling at least one 2.

Two six-sided die are rolled. The figure shows the 36 equally likely outcomes. Find the theoretical probability of rolling at least one 2.

Transcript text: Two six-sided die are rolled. The figure shows the 36 equally likely outcomes. Find the theoretical probability of rolling at least one 2.

Solution

Solution Steps

To find the theoretical probability of rolling at least one 2 with two six-sided dice, we can use the concept of complementary probability. First, calculate the probability of not rolling any 2s, and then subtract this from 1 to get the probability of rolling at least one 2.

Step 1: Total Outcomes

When rolling two six-sided dice, the total number of possible outcomes is given by: \[ \text{Total Outcomes} = 6 \times 6 = 36 \]

Step 2: Non-2 Outcomes

To find the number of outcomes where neither die shows a 2, we consider that each die has 5 options (1, 3, 4, 5, 6). Thus, the number of outcomes without a 2 is: \[ \text{Non-2 Outcomes} = 5 \times 5 = 25 \]

Step 3: Probability Calculations

The probability of not rolling any 2s is calculated as: \[ P(\text{No 2s}) = \frac{\text{Non-2 Outcomes}}{\text{Total Outcomes}} = \frac{25}{36} \approx 0.6944 \] Consequently, the probability of rolling at least one 2 is: \[ P(\text{At least one 2}) = 1 - P(\text{No 2s}) = 1 - \frac{25}{36} = \frac{11}{36} \approx 0.3056 \]