Tree Diagrams (Grade 11 NSC Matric Mathematics): Revision Notes

Worked Example: Weather Prediction Problem

Problem: If it rains today, there's a $\frac{1}{3}$ probability it will rain tomorrow. If it doesn't rain today, there's a $\frac{1}{6}$ probability it will rain tomorrow. The probability of rain today is $\frac{1}{5}$. What's the probability it will rain the day after tomorrow?

Solution:

Step 1: Set up the first level The probability of rain today is $\frac{1}{5}$, so the probability of no rain today is $\frac{4}{5}$.

Step 2: Add the second level

• If it rains today: $P(\text{rain tomorrow}) = \frac{1}{3}$ and $P(\text{no rain tomorrow}) = \frac{2}{3}$
• If no rain today: $P(\text{rain tomorrow}) = \frac{1}{6}$ and $P(\text{no rain tomorrow}) = \frac{5}{6}$

Step 3: Calculate the probability We want rain on the day after tomorrow, which can happen in two ways:

• Path 1: Rain today → Rain tomorrow → Rain day after tomorrow
• Path 2: No rain today → No rain tomorrow → Rain day after tomorrow

For Path 1: $P = \frac{1}{5} \times \frac{1}{3} = \frac{1}{15}$

For Path 2: $P = \frac{4}{5} \times \frac{1}{6} = \frac{2}{15}$

Total probability: $\frac{1}{15} + \frac{2}{15} = \frac{1}{5}$

Worked Example: Coin Flipping Game

Problem: You flip a coin up to 3 times. If you get tails, you earn 2 points and stop. If you get heads, you earn 1 point and flip again. What's the probability of earning exactly 3 points?

Solution:

Step 1: Identify the events Each coin flip has probability $\frac{1}{2}$ for heads (H) and $\frac{1}{2}$ for tails (T).

• Heads = 1 point, continue flipping
• Tails = 2 points, game ends

Step 2: Draw the complete tree We can earn exactly 3 points in two ways:

• H, T (1 + 2 = 3 points)
• H, H, H (1 + 1 + 1 = 3 points)

Step 3: Calculate probabilities

• $P(\text{H, T}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
• $P(\text{H, H, H}) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$

Total probability: $\frac{1}{4} + \frac{1}{8} = \frac{3}{8}$

Worked Example: Medical Trial Analysis

Problem: In a medical trial, half the patients receive medicine and half receive a sugar pill. The medicine cures 60% of patients who take it, while only 10% of patients improve with the sugar pill. What's the probability that a randomly selected patient gets cured?

Solution:

Step 1: Identify the probabilities

• $P(\text{medicine}) = \frac{1}{2}$, $P(\text{sugar pill}) = \frac{1}{2}$
• $P(\text{cured | medicine}) = \frac{3}{5}$, $P(\text{not cured | medicine}) = \frac{2}{5}$
• $P(\text{cured | sugar pill}) = \frac{1}{10}$, $P(\text{not cured | sugar pill}) = \frac{9}{10}$

Step 2: Calculate using the tree diagram Two paths lead to being cured:

• Medicine path: $\frac{1}{2} \times \frac{3}{5} = \frac{3}{10}$
• Sugar pill path: $\frac{1}{2} \times \frac{1}{10} = \frac{1}{20}$

Step 3: Find total probability $P(\text{cured}) = \frac{3}{10} + \frac{1}{20} = \frac{6}{20} + \frac{1}{20} = \frac{7}{20}$