Probability Calculations

This page is aimed to help you with all the basics of probability. You will learn how to list outcomes, how to use Venn diagrams and probability trees. You will also find out about mutually exclusive events and independent events and get plenty of practice in the kind of probability questions that come up in the examinations.


Key Concepts

On this page, you should learn to

  • the probability of an event
  • complementary events
  • expected number of occurrences
  • sample spaces
  • Venn diagrams
  • tree diagrams
  • combined events: OR, AND
  • mutually exclusive events
  • independent events

Summary

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Test Yourself

Here is a quiz that gets you to practise set notation and Venn diagrams


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Here is a quiz about mutually exclusive and independent events 


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Here is a quiz about the probability rule \(P(A\cup B)\) = P(A) + P(B) - \(P(A\cap B)\) . You might find Venn diagrams helpful to answer these questions


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The final quiz on this page is about independent events. You might find tree diagrams helpful to answer these questions


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Exam-style Questions

Question 1

Two events A and B are such that P(A) = 0.35 and P(B) = 0.6 and \(P(A\cup B)\) = 0.74

a) Find \(P(A\cap B)\)

b) Determine whether A and B are independent

c) Find \(P(A'\cap B)\)

Hint

Full Solution

Question 2

Katniss is practising archery. She fires three arrows at a target. The probability that she hits the target with her first arrow is 0.7. Whenever she hits the target, her confidence increases so that the probability that she hits the target on her next attempt increases by 0.1. Whenever she misses the target, the probability reduces by 0.1.

a) Complete the probability tree for Katniss’s three attempts.


b) Calculate the probability that she hits the target with two attempts.

c) Find the probability the she hits the target on at least one attempt.

Hint

Full Solution

Question 3

Events A and B are such that P(A) = 0.4 and \(P(A\cup B)\) = 0.7

Find P(B) if A and B are independent

Hint

Full Solution

Question 4

Alphonse and Bettina are playing a game. A bag contains 2 yellow beads and 3 red beads. They take it in turns to pick a bead from the bag at random. Alphonse goes first. If Alphonse picks a yellow bead, he wins and the game stops. If he picks a red bead, he replaces the bead and it is Bettina’s turn. If Bettina picks a red bead, she wins and the game stops. If she picks a yellow bead, she replaces the bead and it is Alphonse’s turn again.

a) Find the probability that Alphonse wins on his first turn.

b) Show that the probability that Alphonse wins on his second turn is \(\frac{12}{125}\)

c) The game continues until one of the player wins. What is the probability that Alphonse wins the game?

Hint

Full Solution

 

Question 5

A box contains 25 tickets. x tickets are gold, the rest are silver. Two tickets are selected at random.

a) Show that the probability of selecting two gold tickets is \(\frac{x^2-x}{600}\)

b) Find the probability of selecting two tickets of the same colour.

c) The probability of selecting two tickets of the same colour is twice the probability of selecting two tickets of a different colour. Find how many gold tickets there are.

Hint

Full Solution

MY PROGRESS

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