A die is thrown three times. The probability of obtaining at least one six is

Options

A. `125/216`
B. `91/216`
C. `1/216`
D. `215/216`

Answer

To find the probability of obtaining at least one six when a die is thrown three times, we can follow these steps:

### Step 1: Understand the Problem
We need to calculate the probability of getting at least one six in three throws of a die.

**Hint:** Remember that "at least one" means we can have one, two, or three sixes.

### Step 2: Use the Complement Rule
Instead of calculating the probability of getting one, two, or three sixes directly, we can use the complement rule. The complement of getting at least one six is getting no sixes at all.

**Hint:** The complement rule states that P(A) = 1 - P(A'), where A' is the complement of event A.

### Step 3: Calculate the Probability of No Sixes
The probability of not getting a six in a single throw of a die is \( \frac{5}{6} \). If the die is thrown three times, the probability of not getting a six in all three throws is:
\[ P(\text{no sixes}) = \left(\frac{5}{6}\right)^3 \]

**Hint:** When calculating the probability of independent events, multiply the probabilities of each event.

### Step 4: Calculate \( \left(\frac{5}{6}\right)^3 \)
Now, we compute:
\[ \left(\frac{5}{6}\right)^3 = \frac{5^3}{6^3} = \frac{125}{216} \]

**Hint:** Make sure to calculate \( 5^3 \) and \( 6^3 \) correctly.

### Step 5: Use the Complement to Find the Desired Probability
Now, we can find the probability of getting at least one six:
\[ P(\text{at least one six}) = 1 - P(\text{no sixes}) = 1 - \frac{125}{216} \]

**Hint:** Subtract the probability of the complement from 1.

### Step 6: Calculate the Final Probability
Now, we perform the subtraction:
\[ P(\text{at least one six}) = 1 - \frac{125}{216} = \frac{216 - 125}{216} = \frac{91}{216} \]

### Step 7: Convert to Decimal (if needed)
To express this probability as a decimal:
\[ \frac{91}{216} \approx 0.4213 \]

### Final Answer
Thus, the probability of obtaining at least one six when a die is thrown three times is approximately \( 0.421 \).

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