A fair coin is tossed 100 times. What is the probability of getting tails an odd number of times?

Options

A. `(1)/(2)`
B. `(3)/(8)`
C. `(1)/(4)`
D. `(1)/(8) `

Answer

To find the probability of getting tails an odd number of times when a fair coin is tossed 100 times, we can follow these steps:

### Step 1: Understand the Total Outcomes
When a fair coin is tossed, there are two possible outcomes: heads (H) or tails (T). When the coin is tossed 100 times, the total number of outcomes is \(2^{100}\).

**Hint:** Remember that each toss has 2 outcomes, and the total number of tosses determines the exponent.

### Step 2: Define the Random Variable
Let \(X\) be the random variable representing the number of tails obtained in 100 tosses. The possible values of \(X\) range from 0 to 100.

**Hint:** Think about how the number of tails can vary with each toss.

### Step 3: Identify Odd and Even Outcomes
The outcomes can be classified into two categories: odd and even. Since there are 100 tosses, the number of tails can be odd (1, 3, 5, ..., 99) or even (0, 2, 4, ..., 100).

**Hint:** List the odd numbers between 0 and 100 to visualize the distribution.

### Step 4: Use the Binomial Distribution
The number of tails \(X\) follows a binomial distribution with parameters \(n = 100\) (number of trials) and \(p = 0.5\) (probability of getting tails in each trial). The probability of getting exactly \(k\) tails is given by:

\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

**Hint:** Recall the formula for binomial probability and how to calculate combinations.

### Step 5: Calculate the Probability of Odd Outcomes
To find the probability of getting an odd number of tails, we can sum the probabilities of all odd outcomes:

\[ P(\text{odd}) = P(X = 1) + P(X = 3) + P(X = 5) + ... + P(X = 99) \]

However, there's a simpler way to find this probability. The sum of the probabilities of all outcomes (odd and even) must equal 1. Due to symmetry in a fair coin toss, the probability of getting an odd number of tails is equal to the probability of getting an even number of tails.

**Hint:** Think about the symmetry in the distribution of outcomes.

### Step 6: Conclude the Probability
Since the total probability is 1 and the probabilities of odd and even outcomes are equal, we have:

\[ P(\text{odd}) + P(\text{even}) = 1 \]

Thus,

\[ P(\text{odd}) = P(\text{even}) = \frac{1}{2} \]

**Final Answer:** The probability of getting tails an odd number of times when a fair coin is tossed 100 times is \(\frac{1}{2}\).