8.05 Probability using combinations

A box contains 6 pens of different colours: red, green, blue, yellow, black and white. Two pens are drawn at random without replacement.

A newspaper editor is deciding which of 5 articles to print on the front page.

5 people are to be selected from a larger group of 10 candidates. If Amelia is among the candidates, find the probability that she will be among those selected.

5 cards are chosen at random without replacement from a standard deck of 52 cards. Find the probability that they are all red.

A container contains twelve cards. The cards are marked with the numbers 1 to 12. From the container, five cards are drawn at random without replacement.

The nominations for the school captain and vice captain were Tom, Fred, Irene, Lisa and Kate. The school captain is chosen first, then the vice captain.

In a football squad, there are 9 midfielders. 5 midfielders are selected to play in a game.

Companies prefer their product to be placed on the middle shelf in supermarkets. There are 6 brands, but only space for 3 of them on the middle shelf.

Senior students are asked to select 8 subjects from any of 7 mainstream subjects and 6 elective subjects.

4 cards are chosen at random without replacement from a standard deck of 52 cards. Find the following probabilities, as percentages to two decimal places:

Bottles of water are capped by machinery on a production line. After they have been capped, bottles are randomly selected to test that they are correctly sealed. Of the first 20 bottles to come off the production line, 7 of them are not correctly sealed. Georgia randomly selects 5 of the first 20 bottles.

From 8 teachers and 7 students, 4 people are randomly selected.

A drawer contains 18 batteries, of which 4 have low charge and the rest are fully charged. Vanessa needs to choose three of the batteries to operate a remote control. Find the probability that:

An airline wants to allocate 5 flight attendants to a particular flight route. They randomly select them from 8 staff who speak a European language only and 6 staff who speak an Asian language only.

There are 5 red, 5 white and 3 blue marbles in a bag. 5 marbles are chosen at random from the bag, without replacement. Find the probability that:

6 cards are chosen from a standard deck of 52 cards, without replacement.

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.

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.

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:

Four cards of different suits (diamonds, hearts, spades and clubs) are placed in a pile. A card is selected from the pile. Its suit is recorded and the card is returned to the pile. A second card is then chosen and its suit is also recorded.

The nominations for the school captain and vice captain were Ben, Bart, Ellie, Hannah and Emma. The school captain is to be chosen first from these candidates , and then the vice captain. Find the probability that Ben and Ellie will fill the positions.

3 letters of the word ENVIOUS are chosen at random. Find the probability that the selection includes just 1 vowel.

At a film festival, 6 films are to be chosen to go in the running for grand prize of best film. There are 8 foreign films and 7 local films to choose from. If the selection is made randomly, find the probability that 4 foreign films and 2 local films are chosen.

A local grocery store offers 26 different flavours of ice-cream. A cone can hold 2 flavours.

You order 12 sandwiches for your office lunch. 5 are roast beef, 5 are ham and 2 are turkey. The sandwich shop wrapped them but forgot to label which meat was in each sandwich. If you pick 4 sandwiches at random, find the probability of the following as a percentage to two decimal places.

In the Have Sum Fun competition, a teacher needs to make a team of 4 people and another team of 3 people. For the larger team, the teacher has 5 students to choose from, including Jack and Julia. For the smaller team, the teacher has 4 students to choose from, including Alvin. The selections are made independently.

Find the probability both Jack and Alvin are chosen.

Valentina is creating an exercise plan from a list of 30 exercises. Valentina has a total of 10 cardio exercises, 12 gymnastics exercises and 8 weight exercises. If Valentina wants 6 exercises and one of the chosen exercises is cardio, find the probability that all 6 are cardio.

Laura is allowed to pack 4 toys for her weekend trip to Grandma’s house. Laura has 6 dolls, 7 cars and 8 teddy bears.

Roald is taking 4 books to read on his holiday. He has 3 biographies, 6 novels and 5 non-fiction books to choose from.

Find the probability Roald will take at least one biography.

Christa is painting 3 bedrooms in her house. Christa has 6 colours to choose from for the 3 bedrooms.

A tour guide is taking groups on holiday and wants to mix up the nationalities of the groups, but she does this at random. Each group has 6 people. On this tour, there are 10 Australians, 7 Americans and 12 Chinese people.

In one group, find the probability that there are 2 people of each nationality.

A cricket team consists of 11 players, including a captain and a vice-captain. The coach decides on the basis of fairness to select the captain and vice-captain at random.