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Math  /  Data & Statistics

QuestionTen balls numbered from 1 to 10 are placed into a bag. Some are grey and some are white. The balls numbered 1,3,5,7,8,91,3,5,7,8,9, and 10 are grey. The balls numbered 2,4 , and 6 are white. A ball is selected at random. Let XX be the event that the selected ball is white, and let P(X)P(X) be the probability of XX.
Let not XX be the event that the selected ball is not white, and let PP (not XX ) be the probability of not XX. (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} & \multirow[b]{2}{*}{Probability} \\ \hline & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & \\ \hline X & - & 10 & C & 7 & 4 & ( & 0 & 4 & 0 & 0 & P(X)=□P(X)=\square \\ \hline not X & 0 & T & 0 & \% & 4 & (1) & ( & th & \% & 4 & P(not⁡X)=□P(\operatorname{not} X)=\square \\ \hline \end{tabular} (b) Subtract. 1−P(X)=1-P(X)= □\square

Studdy Solution

STEP 1

1. There are 10 balls, each uniquely numbered from 1 to 10.
2. Balls numbered 1, 3, 5, 7, 8, 9, and 10 are grey.
3. Balls numbered 2, 4, and 6 are white.
4. The probability of selecting a ball is uniform across all balls.

STEP 2

1. Identify the outcomes for event XX.
2. Calculate the probability P(X)P(X).
3. Identify the outcomes for event not X\text{not } X.
4. Calculate the probability P(not X)P(\text{not } X).
5. Verify the relationship 1−P(X)1 - P(X).

STEP 3

Identify the outcomes for event XX, which is the event that the selected ball is white. The white balls are numbered 2, 4, and 6.
Outcomes for XX: \{2, 4, 6\}

STEP 4

Calculate the probability P(X)P(X). There are 3 white balls out of 10 total balls.
P(X)=310 P(X) = \frac{3}{10}

STEP 5

Identify the outcomes for event not X\text{not } X, which is the event that the selected ball is not white. The grey balls are numbered 1, 3, 5, 7, 8, 9, and 10.
Outcomes for not X\text{not } X: \{1, 3, 5, 7, 8, 9, 10\}

STEP 6

Calculate the probability P(not X)P(\text{not } X). There are 7 grey balls out of 10 total balls.
P(not X)=710 P(\text{not } X) = \frac{7}{10}

STEP 7

Verify the relationship 1−P(X)1 - P(X).
1−P(X)=1−310=710 1 - P(X) = 1 - \frac{3}{10} = \frac{7}{10}
This confirms that:
P(not X)=710 P(\text{not } X) = \frac{7}{10}
The completed table is:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \multirow{2}{*}{Event} & \multicolumn{10}{|c|}{Outcomes} & \multirow[b]{2}{*}{Probability} \\ \hline & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & \\ \hline X & - & \checkmark & - & \checkmark & - & \checkmark & - & - & - & - & P(X)=\frac{3}{10} \\ \hline \text{not } X & \checkmark & - & \checkmark & - & \checkmark & - & \checkmark & \checkmark & \checkmark & \checkmark & P(\text{not } X)=\frac{7}{10} \\ \hline \end{tabular}
(b) Subtract:
1−P(X)=710 1 - P(X) = \frac{7}{10}

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