Understanding the Coin Toss Game Probability

In this game, we are tossing three fair coins simultaneously. We need to determine the possible outcomes and the probability of each outcome to calculate the expected amount the child will lose.

The rule is simple: win Rs 5 for a specific outcome (all heads or all tails) and lose Rs 3 for any other outcome. We want to find the average amount lost over many games, which is the expected loss.

Analyzing Possible Outcomes of Tossing Three Coins

When three coins are tossed, each coin can land either heads (H) or tails (T). The total number of possible outcomes is \(2 \times 2 \times 2 = 2^3 = 8\). These outcomes form the sample space of the experiment.

The possible outcomes are:

• HHH
• HHT
• HTH
• THH
• HTT
• THT
• TTH
• TTT

Since the coins are fair and the tosses are independent, each of these 8 outcomes is equally likely. The probability of any single outcome occurring is \(\frac{1}{8}\).

Calculating Probabilities of Winning and Losing

The child wins Rs 5 if the outcome is HHH or TTT.

• Number of winning outcomes = 2 (HHH, TTT)
• Probability of winning, \(P(\text{Win})\) = (Number of winning outcomes) / (Total number of outcomes) = \(\frac{2}{8} = \frac{1}{4}\)

The child loses Rs 3 for all other cases.

• Number of losing outcomes = Total outcomes - Winning outcomes = \(8 - 2 = 6\)
• Probability of losing, \(P(\text{Lose})\) = (Number of losing outcomes) / (Total number of outcomes) = \(\frac{6}{8} = \frac{3}{4}\)

Alternatively, \(P(\text{Lose}) = 1 - P(\text{Win}) = 1 - \frac{1}{4} = \frac{3}{4}\).

Calculating Expected Value of the Game

The expected value (\(E\)) of a game is the average outcome if the game were played many times. It is calculated using the formula:

\(E = \sum (\text{Value of Outcome} \times \text{Probability of Outcome})\)

In this game, we have two main outcomes with associated probabilities and values:

• Outcome: Win Rs 5, Probability: \(P(\text{Win}) = \frac{1}{4}\)
• Outcome: Lose Rs 3 (gain of -3), Probability: \(P(\text{Lose}) = \frac{3}{4}\)

Now, let's calculate the expected value:

\(E = (\text{Amount Won} \times P(\text{Win})) + (\text{Amount Lost} \times P(\text{Lose}))\)

\(E = (5 \times \frac{1}{4}) + (-3 \times \frac{3}{4})\)

\(E = \frac{5}{4} - \frac{9}{4}\)

\(E = \frac{5 - 9}{4}\)

\(E = \frac{-4}{4}\)

\(E = -1\)

Conclusion on Expected Loss

The expected value is -1. A negative expected value means that, on average, the child is expected to lose money per game over the long run. An expected value of -1 means the expected outcome is a loss of Rs 1.

Therefore, the expected amount to lose in the game is Rs 1.

Revision Table: Key Concepts in Probability Games

Additional Information: Probability and Expected Value

Probability: Probability is a fundamental concept in statistics that quantifies the likelihood of an event. It is always a number between 0 and 1, inclusive. A probability of 0 means the event is impossible, and a probability of 1 means the event is certain. In our coin toss example, each of the 8 outcomes has a probability of \(\frac{1}{8}\), which is between 0 and 1.

Expected Value: The expected value isn't necessarily an outcome you will get in a single trial, but rather the average outcome over a very large number of trials. In financial or game contexts, a positive expected value suggests that, on average, you can expect to make money in the long run. A negative expected value, as in this coin toss game (\(E = -1\)), suggests that, on average, you are expected to lose money over time. An expected value of zero means the game is fair, with no expected gain or loss.

Understanding expected value is crucial in analyzing games of chance, investments, and decision-making under uncertainty. It helps you assess the average result you can anticipate.