Problem 29 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If a coin be tossed \(n\) times, then probability that the head comes odd number of times is (a) \(1 / 2\) (c) \(1 / 2^{n-1}\) (d) None of these

Step-by-Step Solution

Answer

The probability is \(1/2\), which is option (a).

Step 1: Understanding the Problem

We need to find the probability that when a coin is tossed \(n\) times, heads appear an odd number of times. A coin toss has two outcomes: heads or tails. We are concerned with cases where the number of heads is odd.

Step 2: Analyzing Outcomes Using Binomial Distribution

Each coin toss is an independent event, with heads or tails as outcomes, each having probability \(\frac{1}{2}\). The number of heads in \(n\) tosses follows a binomial distribution \(B(n, \frac{1}{2})\). The probability of getting \(k\) heads is given by \(P(X = k) = \binom{n}{k} \left(\frac{1}{2}\right)^n\).

Step 3: Identifying Odd Number of Heads

For the probability of odd heads, we are interested in the sum of probabilities where \(k\) is odd. That is, \(P(X = 1) + P(X = 3) + ... + P(X = n)\) if \(n\) is odd, or up to \(P(X = n-1)\) if \(n\) is even.

Step 4: Using Symmetry in Binomial Distribution

From the properties of binomial distribution, the outcomes where heads appear are symmetric. Hence, the number of cases with odd heads is equal to the number of cases with even heads. Thus, the probability of getting either is \(\frac{1}{2}\).

Step 5: Conclusion and Answer Verification

Since the probability that a coin shows heads an odd number of times equals the probability it shows heads an even number of times due to symmetry, both are \(\frac{1}{2}\). Thus, the correct choice is \(\frac{1}{2}\), which matches option (a).

Key Concepts

Probability Theory

Probability theory is a mathematical framework used to analyze the likelihood of various outcomes in random phenomena. It helps us predict how often events will occur. For a coin toss, which can end in heads or tails, probability theory provides a way to systematically calculate the chances of each of these outcomes.

• **Sample Space**: In probability theory, the set of all possible outcomes is called the sample space. For a single coin toss, this is {Heads, Tails}.
• **Event**: An event is any subset of the sample space. For example, getting heads is an event.
• **Probability**: The probability of an event is calculated by dividing the number of successful outcomes (favorable to the event) by the total number of outcomes in the sample space. For a fair coin, the probability of getting heads is \( \frac{1}{2} \).

These concepts are crucial for understanding more complex scenarios like the probability of getting a specific number of heads when tossing a coin multiple times.

Odd Number of Outcomes

An odd number of outcomes refers to the particular count of events that results in an odd number, such as 1, 3, or 5 occurrences. In the context of coin tosses, it means counting how many times heads appear in an odd number.
To compute this probability, consider:

• Outcome Set: For 3 tosses, the possible number of times heads can occur ranges from 0 to 3. Here, the odd numbers are 1 and 3.
• Summation of Probabilities: We calculate the individual probabilities of getting these odd counts and add them together. For an experiment with multiple flips, the calculations involve summing probabilities from the binomial distribution formula.

Understanding how to handle odd outcomes is essential because it often involves summation over non-consecutive numbers, requiring recognition of patterns or symmetry in the problem. This ties back to our problem where symmetry in probability across even and odd outcomes simplifies the calculation.

Coin Toss Experiments

Coin toss experiments are classic examples in probability theory used to illustrate basic principles in a straightforward manner. In each toss:

• Outcomes: There are two outcomes: getting heads or tails, each with a probability of \( \frac{1}{2} \).
• Independence: Each toss is independent, meaning the result of one toss does not affect another.

These experiments become interesting as we increase the number of coin tosses. With each additional toss, the number of possible outcomes increases exponentially.

• **Example:** With 2 tosses, there are 4 possible results (HH, HT, TH, TT). With 3 tosses, there are 8 possible outcomes.
• **Binomial Distribution:** This model helps us understand probabilities of combinations of results, like getting exactly 2 heads in 3 tosses.

Coin toss experiments demonstrate fundamental probability concepts, making them popular introductory topics when learning about randomness and distributions.