Scores 1 2 3 4 5 6 Frequency 25 30 x 28 40 32 The table shows the outcome when a die is thrown a number of times. If the probability of obtaining a 3 is 0.2...
Question 1 Report
| Scores | 1 | 2 | 3 | 4 | 5 | 6 |
| Frequency | 25 | 30 | x | 28 | 40 | 32 |
The table shows the outcome when a die is thrown a number of times. If the probability of obtaining a 3 is 0.225;
(a) How many times was the die thrown?
(b) Calculate the probability that a trial chosen at random gives a score of an even number or a prime number.
Answer Details
Given distribution.
| Score | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Frequency | 25 | 30 | x | 28 | 40 | 32 |
(a) Number of throws. Let \(N\) be the total number of throws. Then
\[N = 25+30+x+28+40+32 = 155 + x.\]
The probability of obtaining a 3 is \(\dfrac{x}{N} = 0.225\), so
\[x = 0.225(155 + x) \Rightarrow x = 34.875 + 0.225x \Rightarrow 0.775x = 34.875 \Rightarrow x = 45.\]
Therefore \(N = 155 + 45 = \mathbf{200}\). The die was thrown 200 times.
The completed frequencies are: 25, 30, 45, 28, 40, 32.
(b) P(even number or prime number). Even scores: \(\{2, 4, 6\}\); prime scores: \(\{2, 3, 5\}\). Their union is \(\{2, 3, 4, 5, 6\}\) (only the score 1 is excluded). Adding their frequencies:
\[30 + 45 + 28 + 40 + 32 = 175.\]
\[P(\text{even or prime}) = \frac{175}{200} = \frac{7}{8} = 0.875.\]
Equivalently, \(1 - P(1) = 1 - \dfrac{25}{200} = 0.875\).
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