State the sample space for a two-stage experiment of rolling a standard fair four-sided die, noting whether the number showing is odd or even and then drawing a letter from the wor... State the sample space for a two-stage experiment of rolling a standard fair four-sided die, noting whether the number showing is odd or even and then drawing a letter from the word 'BIB'. Then state the total number of possible events for the experiment.
Understand the Problem
The question asks us to determine the sample space for a two-stage experiment. The first stage involves rolling a four-sided die and noting if the outcome is odd or even. The second stage involves drawing a letter from the word 'BIB'. Finally, the question asks us to calculate the total number of possible outcomes for the experiment.
Answer
$S = \{(Odd, B), (Odd, I), (Even, B), (Even, I)\}$ Total possible outcomes: 4
Answer for screen readers
The sample space is $S = {(Odd, B), (Odd, I), (Even, B), (Even, I)}$.
The total number of possible outcomes is 4.
Steps to Solve
- Determine the possible outcomes of the first stage
The first stage involves rolling a four-sided die. The possible outcomes are the numbers 1, 2, 3, and 4. We are interested in whether the number is odd or even. Therefore, the possible outcomes for the first stage are {Odd, Even}.
- Determine the possible outcomes of the second stage
The second stage involves drawing a letter from the word "BIB". The possible outcomes are {B, I}.
- Determine the sample space
The sample space is the set of all possible outcomes of the two-stage experiment. We can represent it as a set of ordered pairs, where the first element is the outcome of the first stage (Odd or Even) and the second element is the outcome of the second stage (B or I).
The sample space $S$ is: $S = {(Odd, B), (Odd, I), (Even, B), (Even, I)}$
- Calculate the total number of possible outcomes
The total number of possible outcomes is the number of elements in the sample space. In this case, there are 4 elements in the sample space. We could also find this by multiplying the number of outcomes in step 1 and the number of outcomes in step 2: $2 \times 2 = 4$
The sample space is $S = {(Odd, B), (Odd, I), (Even, B), (Even, I)}$.
The total number of possible outcomes is 4.
More Information
The sample space represents all possible combinations of the outcomes from rolling the die (noting odd/even) and drawing a letter from 'BIB'.
Tips
A common mistake is to list the numbers 1, 2, 3, 4 instead of 'Odd' and 'Even' for the first stage. Another common mistake is not listing all the elements of the sample space, or including impossible outcomes.
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