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A local charity holds a carnival to raise money. In one activity, participants make a [tex]$ \$[/tex] 3 [tex]$ donation for a chance to spin a wheel that has 10 spaces with the values 0, 1, 2, 5, and 10. Whatever space it lands on, the participant wins that value. Let $[/tex]X[tex]$ represent the value of a random spin. The distribution is given in the table.

\begin{tabular}{|c|c|c|c|c|c|}
\hline
Value of a Spin & 0 & 1 & 2 & 5 & 10 \\
\hline
Probability & 0.4 & 0.2 & 0.2 & 0.1 & 0.1 \\
\hline
\end{tabular}

Which of the following is the correct interpretation of $[/tex]P(X < 5)$?

A. The probability of a random spin having a value of at most 5 is 0.8.
B. The probability of a random spin having a value of at least 5 is 0.2.
C. The probability of a random spin having a value lower than 5 is 0.8.
D. The probability of a random spin having a value higher than 5 is 0.1.



Answer :

GinnyAnswer
To interpret [tex]\( P(X<5) \)[/tex], we need to understand what the notation means and how to calculate it using the provided distribution table.

1. Definition:
[tex]\( P(X<5) \)[/tex] represents the probability that a random spin will result in a value that is less than 5.

2. Distribution Table:
We should identify all the possible outcomes for [tex]\( X \)[/tex] that are less than 5:
[tex]\[ \begin{array}{|c|c|c|c|} \hline \text{Value of a Spin (X)} & 0 & 1 & 2 & 5 & 10 \\ \hline \text{Probability} & 0.4 & 0.2 & 0.2 & 0.1 & 0.1 \\ \hline \end{array} \][/tex]

Values less than 5 are [tex]\(0, 1, \)[/tex] and [tex]\(2\)[/tex].

3. Summing the Probabilities:
To find [tex]\( P(X<5) \)[/tex]:
[tex]\[ P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) \][/tex]

From the table, we obtain:
[tex]\[ P(X = 0) = 0.4 \][/tex]
[tex]\[ P(X = 1) = 0.2 \][/tex]
[tex]\[ P(X = 2) = 0.2 \][/tex]

4. Calculation:
Adding these probabilities together:
[tex]\[ P(X < 5) = 0.4 + 0.2 + 0.2 = 0.8 \][/tex]

5. Interpretation:
Based on the calculations above, the correct phrase that accurately describes [tex]\( P(X<5) \)[/tex] is:

- The probability of a random spin having a value lower than 5 is 0.8.

Thus, the correct answer to interpret [tex]\( P(X<5) \)[/tex] is:

The probability of a random spin having a value lower than 5 is 0.8.

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