Questions: (a) Eric is on a game show. He will choose a box to see if he wins a prize. The odds in favor of Eric winning a prize are 4/3. Find the probability of Eric winning a prize.

(a) Eric is on a game show. He will choose a box to see if he wins a prize. The odds in favor of Eric winning a prize are 4/3. Find the probability of Eric winning a prize.

Transcript text: (a) Eric is on a game show. He will choose a box to see if he wins a prize. The odds in favor of Eric winning a prize are $\frac{4}{3}$. Find the probability of Eric winning a prize. $\square$

Solution

Solution Steps

To find the probability of Eric winning a prize given the odds in favor of winning are $\frac{4}{3}$, we can use the relationship between odds and probability. The odds in favor of an event are given by the ratio of the probability of the event occurring to the probability of the event not occurring. If the odds in favor are $\frac{4}{3}$, this means the probability of winning (P) to the probability of not winning (1-P) is 4:3. We can set up the equation $\frac{P}{1-P} = \frac{4}{3}$ and solve for P.

Solution Approach

Set up the equation based on the given odds: $\frac{P}{1-P} = \frac{4}{3}$.
Solve for P.

Step 1: Set Up the Equation

Given the odds in favor of Eric winning a prize are $\frac{4}{3}$, we can express this as: \[ \frac{P}{1 - P} = \frac{4}{3} \] where $P$ is the probability of winning.

Step 2: Solve for $P$

To solve for $P$, we rearrange the equation: \[ P = \frac{4}{3} (1 - P) \] \[ P = \frac{4}{3} - \frac{4}{3}P \] \[ P + \frac{4}{3}P = \frac{4}{3} \] \[ P \left(1 + \frac{4}{3}\right) = \frac{4}{3} \] \[ P \left(\frac{7}{3}\right) = \frac{4}{3} \] \[ P = \frac{4}{3} \div \frac{7}{3} \] \[ P = \frac{4}{3} \times \frac{3}{7} \] \[ P = \frac{4}{7} \]

Step 3: Convert to Decimal Form

Convert $\frac{4}{7}$ to a decimal form: \[ P \approx 0.5714 \]