Question

You flip 3 pennies 100 times and get 3 tails out 40 of those times. What is the difference between the experimental probability of getting 3 tails and the theoretical probability? Give answer in simplified fraction.

Answer 1

To determine the difference between the experimental probability and the theoretical probability of getting 3 tails when flipping 3 pennies, we first calculate both probabilities.

**1. Theoretical Probability:**

When flipping a coin, there are two possible outcomes: heads (H) or tails (T). For 3 coins:

- Total outcomes = \(2^3 = 8\)
- The outcomes that result in 3 tails (TTT) is just 1.

Thus, the theoretical probability \(P\) of getting 3 tails is:

\[
P(3 \text{ tails}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{8}
\]

**2. Experimental Probability:**

You flipped the pennies 100 times and obtained 3 tails 40 times. Therefore, the experimental probability \(P_e\) is:

\[
P_e(3 \text{ tails}) = \frac{\text{Number of times 3 tails occurred}}{\text{Total trials}} = \frac{40}{100} = \frac{2}{5}
\]

**3. Difference Between Experimental and Theoretical Probability:**

Now, we need to find the difference between the experimental probability and the theoretical probability:

\[
\text{Difference} = P_e - P = \frac{2}{5} - \frac{1}{8}
\]

To subtract these fractions, we first find a common denominator. The least common multiple of 5 and 8 is 40.

Rewriting the fractions with a common denominator:

\[
\frac{2}{5} = \frac{2 \times 8}{5 \times 8} = \frac{16}{40}
\]

\[
\frac{1}{8} = \frac{1 \times 5}{8 \times 5} = \frac{5}{40}
\]

Now, we can subtract:

\[
P_e - P = \frac{16}{40} - \frac{5}{40} = \frac{16 - 5}{40} = \frac{11}{40}
\]

Thus, the difference between the experimental probability of getting 3 tails and the theoretical probability is:

\[
\boxed{\frac{11}{40}}
\]