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Ten balls numbered from 1 to 10 are placed into a bag. Some are grey and some are white.

The balls numbered [tex]$1, 3, 5, 7, 8, 9, 10$[/tex] are grey. The balls numbered [tex]$2, 4, 6$[/tex] are white.

A ball is selected at random. Let [tex][tex]$X$[/tex][/tex] be the event that the selected ball is white, and let [tex]$P(X)$[/tex] be the probability of [tex]$X$[/tex].

Let [tex]\operatorname{not} X[/tex] be the event that the selected ball is not white, and let [tex]$P(\operatorname{not} X)$[/tex] be the probability of [tex]\operatorname{not} X[/tex].

(a) For each event in the table, check the outcome(s) that are contained in the event. Then, in the last column, enter the probability of the event.

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline \multirow{2}{*}{Event} & \multicolumn{10}{|c|}{Outcomes} \\
\hline & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline [tex]$X$[/tex] & & & & & & & & & & & \\
\hline [tex]$\operatorname{not} X$[/tex] & & & & & & & & & & & \\
\hline
\end{tabular}



Answer :

GinnyAnswer
Sure, let's break down the problem step by step and fill out the table.

First, note the list of outcomes from 1 to 10 and classify them as grey or white:
- Grey balls: [tex]\(1, 3, 5, 7, 8, 9, 10\)[/tex]
- White balls: [tex]\(2, 4, 6\)[/tex]

Let's define the events and compute their probabilities:

### Event [tex]\(X\)[/tex]: The event that the selected ball is white
- Outcomes in event [tex]\(X\)[/tex]: [tex]\(2, 4, 6\)[/tex]
- Total number of balls: [tex]\(10\)[/tex]
- Number of white balls: [tex]\(3\)[/tex]

The probability [tex]\(P(X)\)[/tex] is calculated as:
[tex]\[ P(X) = \frac{\text{Number of white balls}}{\text{Total number of balls}} = \frac{3}{10} = 0.3 \][/tex]

### Event [tex]\(\text{not } X\)[/tex]: The event that the selected ball is not white
- Outcomes in event [tex]\(\text{not } X\)[/tex]: [tex]\(1, 3, 5, 7, 8, 9, 10\)[/tex]
- Number of not white (grey) balls: [tex]\(7\)[/tex]

The probability [tex]\(P(\text{not } X)\)[/tex] is calculated as:
[tex]\[ P(\text{not } X) = \frac{\text{Number of grey balls}}{\text{Total number of balls}} = \frac{7}{10} = 0.7 \][/tex]

Now let's fill in the table.

[tex]\[ \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \multirow{2}{}{ Event } & \multicolumn{10}{|c|}{ Outcomes } & \multirow{2}{}{ Probability } \\ \cline{2-11} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & \\ \hline $X$ & & X & & X & & X & & & & & 0.3 \\ \hline not $X$ & X & & X & & X & & X & X & X & X & 0.7 \\ \hline \end{tabular} \][/tex]

So the final table captures all relevant details:
- Event [tex]\(X\)[/tex] (selecting a white ball) includes outcomes [tex]\(2, 4, 6\)[/tex] with a probability of 0.3.
- Event [tex]\(\text{not } X\)[/tex] (selecting a grey ball) includes outcomes [tex]\(1, 3, 5, 7, 8, 9, 10\)[/tex] with a probability of 0.7.

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