Questions: Counting and Probability Probabilities of an event and its complement 1 2 3 4 5 6 7 8 9 10 ------------------------------------ X P(X)=1/5 not X P(not X)=4/5 (b) Subtract. 1-P(x)=4/5 (c) Select the answer that makes the sentence true. 1-P(X) is the same as (Choose one) P(not X) 1-P(not X) None of the above

Counting and Probability Probabilities of an event and its complement 1 2 3 4 5 6 7 8 9 10 ------------------------------------ X P(X)=1/5 not X P(not X)=4/5 (b) Subtract. 1-P(x)=4/5 (c) Select the answer that makes the sentence true. 1-P(X) is the same as (Choose one) P(not X) 1-P(not X) None of the above
Transcript text: Counting and Probability Probabilities of an event and its complement \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \multicolumn{1}{|c|}{} & \multicolumn{7}{|c|}{ Putcomes } \\ \hline & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & \\ \hline$X$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $P(X)=\frac{1}{5}$ \\ \hline not $X$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $\square$ & $P(\operatorname{not} X)=\frac{4}{5}$ \\ \hline \end{tabular} (b) Subtract. $1-P(x)=\frac{4}{5}$ (c) Select the answer that makes the sentence true. $1-P(X)$ is the same as $\square$ (Choose one) $P(\operatorname{not} X)$ $1-P(\operatorname{not} X)$ None of the above
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Solution

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Solution Steps

Solution Approach
  1. Part (b): To find \(1 - P(X)\), we subtract the probability of event \(X\) from 1. Given \(P(X) = \frac{1}{5}\), we calculate \(1 - \frac{1}{5}\).
  2. Part (c): We need to determine which option is equivalent to \(1 - P(X)\). Since \(1 - P(X)\) is the probability of the complement of \(X\), it should be equal to \(P(\text{not } X)\).
Step 1: Calculate \(1 - P(X)\)

Given that \(P(X) = \frac{1}{5}\), we calculate the complement probability as follows: \[ 1 - P(X) = 1 - \frac{1}{5} = \frac{5}{5} - \frac{1}{5} = \frac{4}{5} \]

Step 2: Determine the Equivalent Expression

We need to determine which option is equivalent to \(1 - P(X)\). Since \(1 - P(X) = \frac{4}{5}\) and \(P(\text{not } X) = \frac{4}{5}\), it follows that: \[ 1 - P(X) = P(\text{not } X) \]

Final Answer

For part (b), the result is: \[ \boxed{1 - P(X) = \frac{4}{5}} \]

For part (c), the correct choice is: \[ \boxed{P(\text{not } X)} \]

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