(a) If a number is chosen at random from the integers 5 to 25 inclusive, find the probability that the number is a multiple of 5 or 3. (b) A bag contains 10...

(a) The integers from 5 to 25 inclusive number \(25 - 5 + 1 = 21\).

• Multiples of 5: 5, 10, 15, 20, 25 \(\rightarrow\) 5 numbers.
• Multiples of 3: 6, 9, 12, 15, 18, 21, 24 \(\rightarrow\) 7 numbers.
• Multiples of both (i.e. of 15): 15 \(\rightarrow\) 1 number.

By inclusion and exclusion, the count of "multiple of 5 or 3" is \(5 + 7 - 1 = 11\).

\[ P(\text{multiple of 5 or 3}) = \frac{11}{21}. \]

(b) 10 balls: 4 blue, 6 red. Picking is with replacement, so each pick has \(P(\text{red}) = \tfrac{6}{10}\) and \(P(\text{blue}) = \tfrac{4}{10}\).

(i) Both red:

\[ \frac{6}{10} \times \frac{6}{10} = \frac{36}{100} = \frac{9}{25}. \]

(ii) Both the same colour (both red or both blue):

\[ \left(\frac{6}{10}\right)^2 + \left(\frac{4}{10}\right)^2 = \frac{36}{100} + \frac{16}{100} = \frac{52}{100} = \frac{13}{25}. \]