Probability Trees

This section explains probability trees and how to use them.A probability tree is a diagram used to show all possible outcomes of a situation. It is particularly useful for calculating the probabilities of combined events, especially when events are sequential (happen one after another) or independent (do not affect each other). Each branch of the tree represents a possible outcome, and the probability of that outcome is written next to the branch.

Key Features of a Probability Tree:

• Branches: Each branch represents a possible outcome of an event.
• Probabilities: The probability of each outcome is written on the branches.
• Multiplying Probabilities: For successive events (one event after another), the total probability of a specific outcome is found by multiplying the probabilities along the branches.

Steps to Create a Probability Tree:

1. Start with the first event: Draw a starting point, then draw branches for each possible outcome, labelling each branch with the probability of that outcome.
2. Add subsequent events: From the end of each branch, draw further branches for each possible outcome of the next event, again labelling the probabilities.
3. Calculate probabilities for combined events: To find the probability of a specific combination of events, multiply the probabilities along the relevant branches.

This video gives you an overview on how to use a probability tree.

Example 1: Simple Probability Tree

Suppose you flip a coin twice. The possible outcomes for each flip are heads (H) or tails (T). The probability of getting heads (H) is 1/2, and the probability of getting tails (T) is also 1/2.

1. First flip: There are two possible outcomes: heads (H) or tails (T), each with a probability of 1/2.
2. Second flip: After the first flip, there are again two possible outcomes for the second flip: heads (H) or tails (T), each with a probability of 1/2.

The probability tree looks like this:

                 Start

                /        \

            H (1/2) T (1/2)

              /  \             /    \

      H (1/2) T (1/2) H (1/2) T (1/2)

To find the probability of getting two heads (HH):

• Follow the branch: H → H
• Multiply the probabilities: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

To find the probability of getting one head and one tail (HT or TH):

• For HT: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
• For TH: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
• Total probability for HT or TH: $\frac{1}{4} + \frac{1}{4} = \frac{1}{2}$

To find the probability of getting two tails (TT):

• Follow the branch: T → T
• Multiply the probabilities: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

Example 2: Probability Tree for Dependent Events

Imagine you have a bag with 5 red balls and 3 blue balls. You pick two balls without replacement. The probability of picking a red ball on the first draw is 58\frac{5}{8}85​, and the probability of picking a blue ball on the first draw is $\frac{3}{8}$. Since the balls are not replaced, the probabilities change after the first draw.

1. First draw: You can pick either a red ball (R) or a blue ball (B).
2. Second draw: After the first draw, the total number of balls decreases by one, so the probabilities change accordingly.

The probability tree looks like this:

                      Start

                     /     \

            R (5/8)         B (3/8)

               / \                    /  \

     R (4/7)  B (3/7)  R (5/7)  B (2/7)

Now, to calculate the probabilities:

• Probability of drawing two red balls (RR):
• Probability of drawing a red ball then a blue ball (RB):
• Probability of drawing a blue ball then a red ball (BR):
• Probability of drawing two blue balls (BB):

$$P(RR)= \frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$$

$$P(RB) = \frac{5}{8} \times \frac{3}{7} = \frac{15}{56}$$

$$P(BR) = \frac{3}{8} \times \frac{5}{7} = \frac{15}{56}$$

$$P(BB) = \frac{3}{8} \times \frac{2}{7} = \frac{6}{56} = \frac{3}{28}$$

Key Points to Remember:

• Independent Events: If events are independent (the outcome of one event doesn't affect the other), simply multiply the probabilities along the branches.
• Dependent Events: If events are dependent (the outcome of the first event affects the probability of the second), adjust the probabilities after each event (e.g., reducing the number of possible outcomes if something is not replaced).
• Multiplying Probabilities: To find the probability of a sequence of events, multiply the probabilities along the branches of the tree.

Probability trees are an effective way to visualise and calculate the probabilities of complex events, whether the events are independent or dependent. Practice drawing and interpreting them to strengthen your understanding!