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...

Answer Details

Given distribution.

(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, \(1 - P(1) = 1 - \dfrac{25}{200} = 0.875\).