EDEXCEL GCSE HIGHER MATHS - What are Probability and Tree Diagrams

Edexcel GCSE Higher Maths: Probability and Tree Diagrams

What is Probability?

Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where:

• 0 represents an event that is impossible.
• 1 represents an event that is certain.
• 0.5 represents an event that is equally likely to occur or not occur.

For example:

• The probability of flipping a coin and getting heads is 0.5.
• The probability of rolling a six on a standard die is 1/6.

Tree Diagrams

Tree diagrams are a useful tool for visualizing and calculating probabilities, especially when dealing with multiple events. They work by:

• Branching out: Each branch represents a possible outcome of an event.
• Labeling: Each branch is labeled with the probability of that outcome.
• Multiplying: To find the probability of a sequence of events, you multiply the probabilities of each branch in the sequence.

Example:

Suppose you have a bag containing 3 red balls and 2 blue balls. You take out one ball at random, then another without replacing the first. We can use a tree diagram to find the probability of getting two red balls:

             / Red (3/5) 
        /-------> / Red (2/4) = 3/10
       /        \ Blue (2/4) = 3/20
  Start -----> 
       \        / Red (3/4) = 3/20
        \-------> \ Blue (1/4) = 1/20
             \ Blue (2/5)

• Step 1: Draw two branches from the starting point, representing the first ball being red or blue. Label them with their respective probabilities (3/5 and 2/5).
• Step 2: From each of these branches, draw two more branches representing the second ball being red or blue. Label them with the updated probabilities (3/4 and 1/4 for red, 2/4 and 1/4 for blue).
• Step 3: To find the probability of getting two red balls, multiply the probabilities along the appropriate branches: (3/5) * (2/4) = 3/10.

Key Points:

• The probabilities on each branch must add up to 1.
• Remember to adjust probabilities for events that happen without replacement.
• Tree diagrams are helpful for visualizing complex scenarios and can make probability calculations easier.

Practice Problems:

1. A bag contains 4 white marbles and 3 black marbles. Two marbles are selected at random, without replacement. Draw a tree diagram to show the possible outcomes and calculate the probability of:

   • Getting two white marbles.
   • Getting one white marble and one black marble.

2. A coin is tossed three times. Draw a tree diagram to show the possible outcomes and calculate the probability of getting:

   • Three heads.
   • At least two heads.

3. A student is taking a test with two multiple-choice questions, each with four possible answers. Draw a tree diagram to show the possible outcomes and calculate the probability of getting:

   • Both answers correct.
   • At least one answer correct.