You are playing roulette at a casino in the United States. The wheel has 18 red slots, 18 black slots, and two green slots. In 4 spins of the wheel what is the probability of at least one red?

To find the probability of at least one red in 4 spins of the roulette wheel, we can use the complement rule. The complement rule states that the probability of an event happening is equal to 1 minus the probability of the event not happening. In this case, the event is "at least one red" in 4 spins. The complement of this event is "no reds" in 4 spins. The probability of no reds in 4 spins can be calculated using the formula for the probability of independent events: \[ P(\text{no reds}) = (1 - P(\text{red}))^4 \] Since there are 18 red slots out of a total of 38 slots (18 red + 18 black + 2 green), the probability of a red slot is: \[ P(\text{red}) = \frac{18}{38} \] Substitute this value into the formula for the probability of no reds: \[ P(\text{no reds}) = \left(1 - \frac{18}{38}\right)^4 \] Now, we can calculate the probability of no reds in 4 spins. After that, we can find the probability of at least one red by subtracting the probability of no reds from 1. Calculate the value by following steps: - step0: Calculate: \(\left(1-\frac{18}{38}\right)^{4}\) - step1: Reduce the fraction: \(\left(1-\frac{9}{19}\right)^{4}\) - step2: Subtract the terms: \(\left(\frac{10}{19}\right)^{4}\) - step3: Use the properties of exponents: \(\frac{10^{4}}{19^{4}}\) - step4: Evaluate the power: \(\frac{10000}{19^{4}}\) The probability of no reds in 4 spins is approximately 0.0767. Now, to find the probability of at least one red, we subtract the probability of no reds from 1: \[ P(\text{at least one red}) = 1 - P(\text{no reds}) \] Substitute the value of \( P(\text{no reds}) \) into the formula: \[ P(\text{at least one red}) = 1 - 0.0767 \] Calculate the probability of at least one red in 4 spins. Calculate the value by following steps: - step0: Calculate: \(1-0.0767\) - step1: Subtract the numbers: \(0.9233\) The probability of at least one red in 4 spins of the roulette wheel is approximately 0.9233 or 92.33%.