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Maths

Probability - Higher

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Conditional probabilities

Conditional probability is when the probability of one event depends on a previous event - eg, if items are being picked from a bag in turn, but not replaced.

A bag contains 4 red marbles and 5 green marbles. If we draw 2 marbles from the bag, then the colour of the second marble will be dependent on the colour of the first.

To work out the probabilities for the first marble:

P (first marble is red) = 4/9

P (first marble is green) = 5/9

Here are two different sets of probabilities. Each set depends on which colour marble was taken out in the first event.

If the first marble out of the bag is red, the probabilities for the second marble are:

P (second marble is red, given that the first marble was red) = 3/8P (second marble is green, given that the first marble was red) = 5/8

However, if the first marble is green, there are four red and four green marbles left in the bag.

P (second marble is red, given that the first marble was green) = 4/8

P (second marble is green, given that the first marble was green) = 4/8

These are known as conditional probabilities. They are best represented on a tree diagram. After each event, the top and bottom numbers of the probability change. In this example, the bottom number will decrease by one each time an event occurs (one marble is removed).

image: tree diagram with conditional probabilities

The easiest way to work out conditional probabilities is to draw a tree diagram with the probabilities on each branch.

To find out the total probability of something which has a couple of possible outcomes, remember to add the probabilities together. Multiply along the branches to find each probability, and add together the outcomes.

Remember that the sum of the probabilities for any set of branches should always be 1.

From the tree diagram, we can see that:

P (both marbles are red) = 4/9 × 3/8 = 12/72 = 1/6

P (one marble of each colour) = 4/9 × 5/8 + 5/9 × 4/8 = 40/72 = 5/9

P (both marbles are green) = 5/9 × 4/8 = 20/72 = 5/18

Question
  • The probability that Alex arrives home on time is 0.7.
  • If he does arrive home on time, the probability that his dinner is burnt is 0.1.
  • If he does not arrive home on time, the probability that his dinner is burnt is 0.8.

What is the probability that Alex arrives home on time and his dinner is not burnt?

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Answer

The probability that Alex arrives home on time and his dinner is not burnt is:

0.7 x 0.9 = 0.63

Question

What is the probability that Alex's dinner is burnt?

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Answer

0.7 x 0.1 + 0.3 x 0.8 = 0.07 + 0.24 = 0.31

three sets of probabilities

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