Year 9: Two-step experiment probability

Back to main page

In Year 9, you’ll learn how to calculate the probability of a series of events happening. These are called "two-step experiments" (or multi-stage experiments). It's crucial to understand that each step’s probability is multiplied by the probability of the next step. We use the concept of ‘conditional probability’ - the probability of event B happening, given that event A has already happened.

Understanding the concept

Let's break it down:

Step 1 Probability: The probability of the first event occurring.
Step 2 Probability: The probability of the second event occurring, *given that* the first event has already happened.
Overall Probability: Multiply the two probabilities together.
Overall Probability = Step 1 Probability * Step 2 Probability

Example 1: Coin and dice

Scenario:

You flip a fair coin. If it lands on heads, you roll a fair six-sided die. What is the probability of rolling a 6?

Step 1: Probability of Heads:

The probability of getting heads on a fair coin is 1/2.

Step 2: Probability of rolling a 6 (given Heads):

If you got heads, the coin flip doesn't affect the die roll. The probability of rolling a 6 on a fair six-sided die is 1/6.

Overall Probability:

Overall Probability = (1/2) * (1/6) = 1/12

Example 2: Drawing cards

Scenario:

You draw two cards from a standard deck of 52 playing cards *without replacement* (meaning you don't put the first card back in).

What is the probability of drawing an Ace on the first draw and a King on the second draw?

Step 1: Probability of Drawing an Ace on the first draw:

There are 4 Aces in a deck of 52 cards. So, the probability is 4/52 = 1/13.

Step 2: Probability of Drawing a King on the second draw (given an Ace was drawn first):

After drawing an Ace, there are now 51 cards left in the deck, and 4 Kings remain. Therefore, the probability of drawing a King is 4/51.

Overall Probability:

Overall Probability = (1/13) * (4/51) = 4/663

Key points

Independent events: The events *must* be independent. The outcome of the first event doesn’t affect the probability of the second.
Without replacement: When drawing items without replacement, the probability changes after each draw.
Clearly define steps: Make sure you clearly define the two steps of the experiment.

Practice with more examples to solidify your understanding of two-step experiment probability!