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Contestants on a gameshow spin a wheel with 24 equally-sized segments. Most of thos...

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Problem

Contestants on a gameshow spin a wheel with 24 equally-sized segments. Most of those segments show different prize amounts, but 2 of them are labeled "bankrupt". Suppose that a contestant is going to spin the wheel twice in a row, and the outcome of one spin doesn't affect the outcome of future spins. What is the probability that NEITHER of the spins land on "bankrupt"? Round your answer to two decimal places.

Answer

0.84

Step-by-step solution

  1. Identify the total number of segments and bankrupt segments: The wheel has 24 equally-sized segments. 2 of these segments are labeled "bankrupt".
  1. Calculate the probability of NOT landing on "bankrupt" in a single spin: The number of segments that are NOT "bankrupt" is $24 - 2 = 22$. The probability of not landing on "bankrupt" in a single spin is the number of non-bankrupt segments divided by the total number of segments: $P(\text{not bankrupt}) = \frac{22}{24}$.
  1. Simplify the probability of not landing on "bankrupt": The fraction $\frac{22}{24}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2: $P(\text{not bankrupt}) = \frac{22 \div 2}{24 \div 2} = \frac{11}{12}$.

4. Calculate the probability of NOT landing on "bankrupt" in two consecutive spins: Since the outcome of one spin does not affect the outcome of future spins, the two spins are independent events. The probability of two independent events both occurring is the product of their individual probabilities. Therefore, the probability that NEITHER of the two spins land on "bankrupt" is: $P(\text{not bankrupt on 1st spin AND not bankrupt on 2nd spin}) = P(\text{not bankrupt on 1st spin}) \times P(\text{not bankrupt on 2nd spin})$.
$$P(\text{neither bankrupt}) = \frac{11}{12} \times \frac{11}{12} = \frac{121}{144}$$

  1. Convert the fraction to a decimal and round: To round the answer to two decimal places, we convert the fraction $\frac{121}{144}$ to a decimal: $121 \div 144 \approx 0.840277...$. Rounding to two decimal places, we get $0.84$.