Problem 22 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

A coin is tossed twice. The probability of getting head both the times is (a) \(1 / 2\) (b) \(1 / 4\) (c) \(3 / 4\) (d) 1

Step-by-Step Solution

Answer

The probability is \(\frac{1}{4}\), so the answer is (b).

Step 1: Understand the Problem

We need to find the probability of getting heads on both coin tosses. A fair coin has a probability of 1/2 for landing on heads for each toss.

Step 2: Calculate Probability for Single Event

Since the coin is tossed twice, we need to find the probability of heads on one toss, which is:\[ P( ext{Head on one toss}) = \frac{1}{2} \]

Step 3: Calculate Combined Probability

To find the probability of two independent events both occurring (getting heads on both tosses), multiply their individual probabilities:\[ P( ext{Head on both tosses}) = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) \]

Step 4: Solve the Multiplication

Multiply the probabilities from Step 3 to find the overall probability:\[ P( ext{Head on both tosses}) = \frac{1}{4} \]

Step 5: Choose the Correct Option

The computed probability (\(\frac{1}{4}\)) matches option (b).

Key Concepts

Understanding Independent Events

In probability, independent events are occurrences where the outcome of one event does not affect the outcome of another. To illustrate this, think about flipping a coin. Each flip is an independent event. What you get on the first flip, heads or tails, doesn't influence what you will get on the second flip.

Let's consider tossing a coin twice. Whether you get a head or tail on the first toss, the chances of getting a head or tail on the second toss remain unaffected. This is what makes them independent events. The probability remains constant at each toss.

• The probability of getting a head is 1 out of 2, or \(\frac{1}{2}\), on each toss.
• The probability of getting a tail is also \(\frac{1}{2}\).

Therefore, no prior toss affects the probability of future tosses.

Basics of a Coin Toss

A coin toss is one of the simplest forms of a random event used to understand probability. It is often used in probability problems because it's a clear example of an event with two possible outcomes.

A fair coin has two sides: heads and tails. Therefore, when you flip a fair coin, the probability of it landing on heads is \(\frac{1}{2}\), and similarly, the probability of it landing on tails is also \(\frac{1}{2}\). These probabilities are equal because the coin is unbiased and symmetrical.

• If you flip a coin once, there are only two possible outcomes.
• This simplicity makes it ideal for learning about basic probability.

This straightforwardness of a single coin toss makes it a perfect starting point to explore more complex probability concepts like combined or multiple event probabilities.

Calculating Combined Probability

In probability, when determining the chance of two independent events both occurring, we calculate what's known as combined probability. This involves multiplying the probability of each event.

Using our earlier example of coin tosses, let's see how this works when tossing a coin twice, looking for heads both times. Each toss is an independent event with a probability of \(\frac{1}{2}\) of getting heads. To find the probability of getting heads on both tosses, you multiply these probabilities together:

\[ P(\text{Head on both tosses}) = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) = \frac{1}{4} \]

• This multiplication reflects the fact that each toss is an independent event.
• The combined probability of getting heads on two consecutive tosses is therefore \(\frac{1}{4}\).

Understanding combined probability is essential as it allows us to calculate the likelihood of complex events, providing a clear picture of possible outcomes when multiple events are involved.