Analyzing Coin Toss Probabilities

We are given the probabilities for the four joint events when tossing two coins, R and S:

• $P({H_R}{H_S}) = 0.28$
• $P({T_R}{T_S}) = 0.18$
• $P({H_R}{T_S}) = 0.30$
• $P({T_R}{H_S}) = 0.24$

First, let's check if the sum of these probabilities is 1:

$0.28 + 0.18 + 0.30 + 0.24 = 1.00$. The probabilities are consistent.

Calculating Marginal Probabilities for Coins R and S

To determine if the coins are fair or if the tosses are independent, we need to calculate the marginal probabilities for each outcome (Head or Tail) for each coin.

Probability for Coin R

The probability of getting a Head for Coin R ($P(H_R)$) is the sum of probabilities where R shows Head:

$P(H_R) = P({H_R}{H_S}) + P({H_R}{T_S}) = 0.28 + 0.30 = 0.58$

The probability of getting a Tail for Coin R ($P(T_R)$) is the sum of probabilities where R shows Tail:

$P(T_R) = P({T_R}{T_S}) + P({T_R}{H_S}) = 0.18 + 0.24 = 0.42$

Check: $P(H_R) + P(T_R) = 0.58 + 0.42 = 1.00$.

Probability for Coin S

The probability of getting a Head for Coin S ($P(H_S)$) is the sum of probabilities where S shows Head:

$P(H_S) = P({H_R}{H_S}) + P({T_R}{H_S}) = 0.28 + 0.24 = 0.52$

The probability of getting a Tail for Coin S ($P(T_S)$) is the sum of probabilities where S shows Tail:

$P(T_S) = P({H_R}{T_S}) + P({T_R}{T_S}) = 0.30 + 0.18 = 0.48$

Check: $P(H_S) + P(T_S) = 0.52 + 0.48 = 1.00$.

Checking for Independence in Coin Tosses

Two events are independent if the probability of both occurring is equal to the product of their individual probabilities. That is, for any events A and B, independence holds if $P(A \cap B) = P(A) \times P(B)$.

Let's test this condition for the joint event ${H_R}{H_S}$:

The given joint probability is $P({H_R}{H_S}) = 0.28$.

The product of the individual probabilities is $P(H_R) \times P(H_S) = 0.58 \times 0.52 = 0.3016$.

Comparing the two values:

$0.28 \neq 0.3016$.

Since the condition $P({H_R}{H_S}) = P(H_R) \times P(H_S)$ is not met, the coin tosses are dependent.

Assessing Coin Fairness

A coin is considered fair if the probability of getting a Head is $0.5$ and the probability of getting a Tail is $0.5$.

Fairness of Coin R

For Coin R, $P(H_R) = 0.58$ and $P(T_R) = 0.42$. Since neither probability is $0.5$, Coin R is not fair.

Fairness of Coin S

For Coin S, $P(H_S) = 0.52$ and $P(T_S) = 0.48$. Since neither probability is $0.5$, Coin S is not fair.

Evaluating the Given Options

Based on our analysis:

• Option 1: The coin tosses are independent - This is FALSE because $P({H_R}{H_S}) \neq P(H_R) \times P(H_S)$.
• Option 2: R is fair, S is not. - This is FALSE because Coin R is not fair ($P(H_R) = 0.58$).
• Option 3: S is fair, R is not. - This is FALSE because Coin S is not fair ($P(H_S) = 0.52$).
• Option 4: The coin tosses are dependent - This is TRUE because the condition for independence was violated.

Therefore, the correct statement is that the coin tosses are dependent.