A die is thrown 7 times. The chance that an odd number turns up at least 4 times, is

Options

A. `1//4`
B. `1//2`
C. `1//8`
D. None of these

Answer

To solve the problem of finding the probability that an odd number turns up at least 4 times when a die is thrown 7 times, we can follow these steps:

### Step 1: Identify the Probability of Getting an Odd Number
A standard die has 6 faces: 1, 2, 3, 4, 5, and 6. The odd numbers are 1, 3, and 5. Thus, there are 3 odd numbers out of 6 total numbers.

**Probability of rolling an odd number (P(Odd))**:
\[ P(Odd) = \frac{3}{6} = \frac{1}{2} \]

### Step 2: Identify the Probability of Getting an Even Number
Similarly, the even numbers on a die are 2, 4, and 6, which also gives us:

**Probability of rolling an even number (P(Even))**:
\[ P(Even) = \frac{3}{6} = \frac{1}{2} \]

### Step 3: Define the Random Variable
Let \( X \) be the random variable representing the number of times an odd number appears when the die is thrown 7 times. We want to find \( P(X \geq 4) \).

### Step 4: Use the Complement Rule
To find \( P(X \geq 4) \), we can use the complement rule:
\[ P(X \geq 4) = 1 - P(X < 4) \]
This means we need to calculate \( P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \).

### Step 5: Calculate Each Probability Using the Binomial Formula
The probability of getting exactly \( k \) odd numbers in \( n \) trials (where \( n = 7 \)) can be calculated using the binomial probability formula:
\[ P(X = k) = \binom{n}{k} (p)^k (1-p)^{n-k} \]
where \( p = \frac{1}{2} \).

#### Calculate \( P(X = 0) \):
\[ P(X = 0) = \binom{7}{0} \left(\frac{1}{2}\right)^0 \left(\frac{1}{2}\right)^{7} = 1 \cdot 1 \cdot \frac{1}{128} = \frac{1}{128} \]

#### Calculate \( P(X = 1) \):
\[ P(X = 1) = \binom{7}{1} \left(\frac{1}{2}\right)^1 \left(\frac{1}{2}\right)^{6} = 7 \cdot \frac{1}{2} \cdot \frac{1}{64} = \frac{7}{128} \]

#### Calculate \( P(X = 2) \):
\[ P(X = 2) = \binom{7}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^{5} = 21 \cdot \frac{1}{4} \cdot \frac{1}{32} = \frac{21}{128} \]

#### Calculate \( P(X = 3) \):
\[ P(X = 3) = \binom{7}{3} \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^{4} = 35 \cdot \frac{1}{8} \cdot \frac{1}{16} = \frac{35}{128} \]

### Step 6: Sum the Probabilities
Now we sum these probabilities:
\[ P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \]
\[ P(X < 4) = \frac{1}{128} + \frac{7}{128} + \frac{21}{128} + \frac{35}{128} = \frac{64}{128} = \frac{1}{2} \]

### Step 7: Calculate \( P(X \geq 4) \)
Now we can find \( P(X \geq 4) \):
\[ P(X \geq 4) = 1 - P(X < 4) = 1 - \frac{1}{2} = \frac{1}{2} \]

### Final Answer
The probability that an odd number turns up at least 4 times when a die is thrown 7 times is:
\[ \frac{1}{2} \]