10.06 Conditional probability

A group of people were asked what type of exercise they like to do. The results are shown in the Venn diagram below:

A person is chosen from the group at ramdom.

a

Find the probability that the person walks.

b

Find the probability that the person runs, given that they also walk.

Two sets of numbers, A and B, are such that set A contains the even numbers from 1 to 20, inclusive, and set B contains the factors of 48 from 1 to 20, inclusive.

At a university there are 816 students studying first year engineering, 497 of whom are female (set F). Of the 348 students studying Civil Engineering (set C), 237 of them are women.

a

Find the value of:

i

w

ii

x

iii

y

iv

z

b

Find the probability that a randomly selected male student does not study Civil Engineering.

A student is choosing two units to study at university: a language and a science unit. They have 4 languages and 7 science units to choose from.

The following spinner is spun and a normal six-sided die is rolled. The product of their respective results is recorded.

a

Construct a table to represent all the possible outcomes.

b

Find the probability that the product was a multiple of 4 given that a 2 was spun.

c

Find the probability that a 6 was rolled given that the product was greater than 10.

d

Find the probability that the product was greater than 4 given that the same number appeared on the dice and the spinner.

Two cards are randomly drawn without replacement from a deck of cards numbered from 1 to 20. Find the probability that the second card is:

Two dice are rolled, one after another. Find the probability, in simplest form, of rolling:

Four identical balls labelled 1, 2, 3 and 4 are in a bag. Two balls are randomly drawn from the bag in succession and without replacement. Find the probability, in simplest form, that:

For breakfast each morning, Marge eats porridge, toast or cereal. With that she will either drink orange juice, tea or hot chocolate. Find the probability that Marge will:

The following two spinners are spun and the sum of their respective spins is recorded:

a

Construct a table to represent all possible outcomes.

b

Find probability that a 7 was spun given that the sum is less than 12.

c

Find the probability that the sum is odd given that a 4 was spun.

Neil watches three episodes of TV each night. He begins with News or Current Affairs, then the next two shows are either Comedy, Horror or Animation. He never watches more than one Horror show, and if he watched the News, he will follow this immediately with a Comedy.

There are two positions available at a company and the applicants have been shortlisted down to 6: Patricia, Nadia, Amy, Aaron, Lachlan, Jimmy. The two people to fill the position will be picked randomly.

Find P \left( A \cap B \right) given the following probabilities for an experiment in which A and B are two possible events.

P \left( \left. B \right| A \right) = 0.8 \text{ and } P \left( A \right) = 0.5

In an experiment, there are only 2 possible outcomes, A and B. If outcome A occurs, outcome B does not occur and vice versa. State whether the following statements are true:

For each of the following Venn diagrams. find:

a

b

In a group of 69 people, 45 said they had been to Europe \left(E\right), 26 said they had been to Asia \left(A\right) and 8 people said they had been to neither Europe nor Asia. Find:

Shoppers were surveyed about how much they spend on groceries each week. Of the total surveyed, 426 spent less than \$200 each week (set A), 824 spent less than \$400 each week (set B) and 154 spent over \$400.

a

Find the value of:

i

x

ii

y

iii

z

b

Find the probability a shopper spent less than \$400 but more than \$200.

c

Find the probability a shopper spent over \$400 if it is known they spent over \$200.

In a survey, 1000 people were asked about whether they buy (set B) or grow (set G) their own herbs. The results are listed below:

• 4\% said they do neither.

• 40\% said they do both.

• Of those who bought their herbs, the probability that they also grew herbs was \dfrac{2}{3}.

This information is to be displayed in the following Venn diagram, where x, y and z represent numbers of people:

a

Find the value of: x and y.

i

x

ii

y

iii

z

b

Find the number of people who only grew their own herbs.

c

Of those who grew herbs, what proportion also bought herbs?

A card is randomly drawn from a standard 52 pack of cards. Find the probability that it's a jack, queen, king or ace if:

A basketball team has a probability of 0.8 of winning its first season and 0.15 of winning its first season and its second season. Find the probability of winning the second season, given they won first.

Consider P \left( A \right) = 0.2 and P \left( B \right) = 0.5:

A netball coach is choosing players for the Goal Keeper and Goal Defence positions out of the following people:

• Goal Keeper position: Beth, Amy, Joy, Tara.

• Goal Defence position: Eve, Cara, Daisy, Kim, Liz.

The selection for each position is made independently.

Two events A and B are such that P \left( A \cap B \right) = 0.1 and P \left( A \right) = 0.5. Calculate P \left( B \right) if events A and B are independent.

Two events A and B are such that P \left( A \cap B \right) = 0.02 and P \left( A \right) = 0.2.

Consider P \left( A \cup B \right) = 0.72, P \left( A \right) = 0.6 and P \left( B \right) = 0.3.

Consider P \left( A \cup B \right) = 0.72, P \left( A \cap B \right) = 0.18 and P \left( B|A \right) = 0.6.

Consider P \left( \left. A \right| B \right) = 0.8 and P \left( A \cup B \right) \rq = 0.15, where A and B are independent:

In a survey, 531 people are asked whether they watch My Kitchen Rules (MKR) or Masterchef (MC). Of the total respondents, 177 people watch both and 65 watch neither. The number who watch MKR is twice the number who watch both.

The following table describes the departures of trains out of a train station for the months of March and April:

The following table has been drawn up to show the results of a survey on smoking:

Buzz surveyed all the students in Year 12 at his school and summarised the results in the following table:

Some students were asked if they are left or right handed. The results are provided in the given table:

In a recent study of automobile accidents, the number of passengers and whether the car rolled over was collected:

Find the probability that:

In a survey, 74 people were asked their age and how many hours they watch television each week. The results are displayed in the following table:

A person is picked at random from the group. Find the probability that:

A test is available to determine whether someone has a particular condition. However it sometimes gives false readings. A false positive is when the person does not have the condition but the test comes back positive, and a false negative is when the person does have the condition but the test comes back negative. A sample of people and their results is presented in the table:

Find the probability that:

In a survey, 200 people were questioned about whether they voted for Labor, Liberal or Greens last election and who they’ll vote for next election. The rows show who people voted for last election and the columns shows who they'll vote for next election:

Of the 100 students starting kindergarten surveyed, 80\% believe in Santa, 41\% believe in the Tooth Fairy and 31\% believe in both.

In a survey, 200 people were questioned as to whether they read novels in paperback form or on an e-reader:

• 30\% said they do both.

• 40\% said they read paperbacks.

• The probability that a person who didn’t read paperbacks also didn’t read on an e-reader was 30\%.

In a survey, 166 children were asked whether they had read or watched at least one of the stories in the Harry Potter series:

• 62 hadn’t read any of the books

• 39 had read and watched at least one of the stories

• 97 had watched the movie

• 4 had never read or watched any of these

In a population, 15\% of people have brown eyes, 25\% of people have blonde hair, and 10\% of people have both brown eyes and blonde hair. A person is chosen randomly from the population. Find the probability that they have:

A flight departs from Melbourne to Sydney. The following probabilities were found with regards to departure times and weather:

• The probability that the flight departs on time, given the weather is fine in Melbourne is 0.8.

• The probability that the flight departs on time, given the weather is not fine in Melbourne is 0.6.

• The probability that the weather is fine on any particular day in July is 0.3.

Find the probability that:

In a survey, 87 people are questioned about whether they own a tablet \left( T \right) or a smartphone\left( S \right). The following probabilities were determined from the results: