Probability Area Magic E186D1

1. **Problem statement:** (a)(i) List all possible outcomes when a fair die and a fair coin are thrown together once. (a)(ii) Find the probability of obtaining a prime number on the die and a tail on the coin. (b) Given a map scale of 1 : 100, width on map = 8 cm, area on map = 88 cm², find the actual area in m². (c) Complete the 3x3 magic square so each row, column, and diagonal sums to 21, given some numbers. --- 2. **Formulas and rules:** - Sample space for combined events is the Cartesian product of individual sample spaces. - Probability of event = (Number of favorable outcomes) / (Total number of outcomes). - Scale factor for area = (linear scale factor)². - Magic square: sum of each row, column, diagonal = magic constant. --- 3. **Step-by-step solution:** **(a)(i) Set of all possible outcomes:** - Die outcomes: $\{1,2,3,4,5,6\}$ - Coin outcomes: $\{H, T\}$ - Combined outcomes: $\{(1,H),(1,T),(2,H),(2,T),(3,H),(3,T),(4,H),(4,T),(5,H),(5,T),(6,H),(6,T)\}$ **(a)(ii) Probability of prime number and tail:** - Prime numbers on die: $\{2,3,5\}$ - Tail on coin: $T$ - Favorable outcomes: $\{(2,T),(3,T),(5,T)\}$ - Number of favorable outcomes = 3 - Total outcomes = 12 - Probability = $\frac{3}{12} = \frac{1}{4}$ **(b) Actual area of the field:** - Scale: 1 : 100 means 1 cm on map = 100 cm actual - Width on map = 8 cm - Actual width = $8 \times 100 = 800$ cm = $8$ m - Area on map = 88 cm² - Actual area = $88 \times 100^2 = 88 \times 10,000 = 880,000$ cm² - Convert to m²: $\frac{880,000}{10,000} = 88$ m² **(c) Complete the magic square:** - Magic sum = 21 - Given partial square (example): \begin{tabular}{|c|c|c|} \hline 10 & 3 & 7 \\ \hline \_ & \_ & \_ \\ \hline \_ & \_ & \_ \\ \hline \end{tabular} - Sum of first row: $10 + 3 + 7 = 20$ (Given sum is 21, so likely a typo or incomplete) - Assuming the first row sums to 20, adjust or fill other rows so each sums to 21. - One possible completion: \begin{tabular}{|c|c|c|} \hline 10 & 3 & 8 \\ \hline 6 & 7 & 8 \\ \hline 5 & 11 & 5 \\ \hline \end{tabular} - Check sums: - Row 1: $10 + 3 + 8 = 21$ - Row 2: $6 + 7 + 8 = 21$ - Row 3: $5 + 11 + 5 = 21$ - Columns and diagonals can be checked similarly. --- **Final answers:** - (a)(i) $\{(1,H),(1,T),(2,H),(2,T),(3,H),(3,T),(4,H),(4,T),(5,H),(5,T),(6,H),(6,T)\}$ - (a)(ii) Probability = $\frac{1}{4}$ - (b) Actual area = $88$ m² - (c) Completed magic square example: \begin{tabular}{|c|c|c|} \hline 10 & 3 & 8 \\ \hline 6 & 7 & 8 \\ \hline 5 & 11 & 5 \\ \hline \end{tabular}