Today, we’re going to learn about finding conditional probabilities using two-way frequency tables. This is a key concept in Probability and Statistics and will help you understand how outcomes relate to each other.

What are Conditional Probabilities?

Conditional probabilities are the probabilities of an event happening, given that another event has already occurred. Sounds a bit complex, doesn’t it? But don’t worry, let’s break it down with a simple example!

Suppose you’re playing a game where you draw a card from a standard deck of 52 cards. Normally, the probability of drawing an Ace would be 4/52, or 1/13. But what if you knew the card was a Heart? Now, there are only 13 Hearts in the deck, so the probability of drawing an Ace becomes 1/13. This is a conditional probability.

Two-Way Frequency Tables

Two-way frequency tables provide a way of representing how two categories relate to each other. They show the frequency of different outcomes, which we can then use to calculate probabilities.

Suppose we conducted a survey of 100 students to find out if they prefer maths or English and whether they are boys or girls. The two-way frequency table might look something like this:

| | Maths | English | Total |

|——-|——-|———|——-|

| Boys | 30 | 20 | 50 |

| Girls | 20 | 30 | 50 |

| Total | 50 | 50 | 100 |

Calculating Conditional Probabilities

To calculate conditional probabilities from a two-way frequency table, you need to remember this formula:

P(A|B) = \frac{P(A \cap B)}{P(B)}

P(A|B) is the probability of event A happening given that event B has occurred. P(A ∩ B) is the probability of both events A and B happening. P(B) is the probability of event B happening.

From the table above, if we wanted to find the probability that a student prefers maths given that they are a boy (P(Maths|Boy)), we would divide the number of boys who prefer maths (30) by the total number of boys (50). So:

P(Maths|Boy) = \frac{30}{50} = 0.6

Practice Questions

Now, let’s tackle some questions!

Easy Level

  1. Using the table above, what is the probability that a student is a girl given that they prefer English?
  2. What is the probability that a student prefers maths given that they are a girl?
  3. What is the probability that a student is a boy given that they prefer maths?
  4. What is the probability that a student prefers English given that they are a boy?
  5. Using the table above, what is the probability that a student is a girl given that they prefer maths?

Medium Level

  1. In a survey of 200 people, 100 liked chocolate, 50 liked vanilla, and 50 liked both. If a person is selected at random, find the probability they like chocolate given they like vanilla.
  2. In a school of 500 students, 200 are in the football team, 100 are in the cricket team, and 50 are in both teams. If a student is selected at random, find the probability they are in the football team given they are in the cricket team.
  3. In a city of 1000 people, 500 use public transport, 300 use personal cars, and 200 use both. If a person is selected at random, find the probability they use public transport given they use a personal car.
  4. In a group of 150 students, 75 like maths, 50 like English, and 25 like both. If a student is selected at random, find the probability they like maths given they like English.
  5. In a survey of 250 people, 100 have a dog, 50 have a cat, and 25 have both. If a person is selected at random, find the probability they have a dog given they have a cat.

Hard Level

  1. In a group of 300 people, 100 smoke, 60 drink, and 40 do both. If a person is selected at random, find the probability they smoke given they drink.
  2. In a school of 600 students, 200 play basketball, 150 play football, and 50 play both. If a student is selected at random, find the probability they play basketball given they play football.
  3. In a city of 2000 people, 800 use an iPhone, 400 use an Android phone, and 200 use both. If a person is selected at random, find the probability they use an iPhone given they use an Android phone.
  4. In a group of 350 people, 150 are overweight, 100 are obese, and 50 are both. If a person is selected at random, find the probability they are overweight given they are obese.
  5. In a group of 400 people, 200 are vegetarian, 100 are vegan, and 50 are both. If a person is selected at random, find the probability they are vegetarian given they are vegan.

The answers to these questions with explanations are listed at the end. Keep practicing and you’ll soon become a pro at this! Happy learning!

Answers

Easy Level

  1. P(Girl|English) = 30/50 = 0.6
  2. P(Maths|Girl) = 20/50 = 0.4
  3. P(Boy|Maths) = 30/50 = 0.6
  4. P(English|Boy) = 20/50 = 0.4
  5. P(Girl|Maths) = 20/50 = 0.4

Medium Level

  1. P(Chocolate|Vanilla) = 50/100 = 0.5
  2. P(Football|Cricket) = 50/100 = 0.5
  3. P(Public Transport|Car) = 200/300 = 0.67
  4. P(Maths|English) = 25/50 = 0.5
  5. P(Dog|Cat) = 25/50 = 0.5

Hard Level

  1. P(Smoke|Drink) = 40/60 = 0.67
  2. P(Basketball|Football) = 50/150 = 0.33
  3. P(iPhone|Android) = 200/400 = 0.5
  4. P(Overweight|Obese) = 50/100 = 0.5
  5. P(Vegetarian|Vegan) = 50/100 = 0.5