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Level 10

8.02 Dependent events and replacement

Worksheet
Dependent and independent events
1

Identify whether the following events are independent or dependent:

a

A person runs a marathon, and then falls ill from exhaustion.

b

A card is randomly selected from a normal deck of cards, and then returned to the deck. The deck is shuffled and another card is selected.

c

Each student is allowed to randomly pick an item from the teacher's prize bag.

2

The probability of two independent events, A and B are, P \left( A \right) = 0.5 and P \left( B \right) = 0.8. Determine the probability of:

a

Both A and B occurring.

b

Neither A nor B.

c

A or B or both.

d

B but not A.

e

A given that B occurs.

3

A standard six-sided die is rolled 691 times.

a

If it lands on a six 100 times, find the probability that the next roll will land on a six.

b

State whether the outcome of the next roll is independent of or dependent on the outcomes of previous rolls.

4

Ursula takes a bus to the station and then immediately gets on a train to work. Is the probability of her missing the train independent or dependent on her missing her bus?

5

On a roulette table, a ball can land on one of 18 red or 18 black numbers.

a

If it lands on a red number on the first go, find the probability that it will land on a red number on the second go.

b

Are the successive events of twice landing on a red number dependent or independent?

6

From a standard pack of cards, 1 card is randomly drawn and then put back into the pack. A second card is then drawn.

a

Find the probability that neither of the cards are spades.

b

Find the probability that at least 1 of the cards is a spade.

7

Mae deals two cards from a normal deck of cards. Calculate the probability that she deals:

a

Two 10s.

b

Two red cards.

c

Two diamonds.

d

A 10 of spades and an Ace of diamonds, in that order.

e

A 10 of clubs and an Ace of spades, in any order.

8

Two standard die are rolled. One is red and one is white. Calculate the probability that:

a

The same number is rolled.

b

The sum of the two outcomes exceeds 9.

c

The red die is 4 and the white die 6.

d

The red die is even and the white die is odd.

e

The sum of the two outcomes is less than 2.

9

Two events A and B are such that P \left( A \cap B \right) = 0.3 and P \left( A \right) = 0.5.

Calculate P \left( B \right) if events A and B are independent.

10

For the following Venn diagrams:

i

Calculate the value of x.

ii

State whether the events A and B are mutually exclusive.

a
b
11

Suppose P \left( A \right) = 0.2 and P \left( B \right) = 0.3.

a

Find the maximum possible value of P \left( A \cup B \right).

b

Identify events A and B as mutually exclusive or independent.

c

Find the minimum possible value of P \left( A \cup B \right).

12

Two events A and B are such that: P \left( A \cap B \right) = 0.04 and P \left( A \right) = 0.4.

a

If P \left( B \right) = 0.1, state whether the events A and B independent.

b

State whether the events A and B mutually exclusive.

13

Suppose P \left( A \cup B \right) = 0.4, P \left( A' \right) = 0.7 and P \left( B' \right) = 0.8.

a

Find P \left( A \cap B' \right).

b

Find P \left( B \cap A' \right).

c

Determine P \left( A \cap B \right).

d

State whether events A and B are mutually exclusive.

With or without replacement
14

Find the probability of drawing a green counter from a bag of 9 green counters and 6 black counters, replacing it and drawing another green counter.

15

A number game uses a basket with 8 balls, all labelled with numbers from 1 to 8. Three balls are drawn at random, with replacement.

Find the probability that the ball labelled 4 is picked:

a

Exactly once

b

Exactly twice

c

Exactly three times.

16

Eileen randomly selects two cards, with replacement, from a normal deck of cards. Calculate the probability that both cards are:

a

Red

b

The same colour.

c

Different colours.

17

Christa randomly selects two cards, with replacement, from a normal deck of cards.

Calculate the probability that:

a

The first card is a Queen of Diamonds and the second card is a 10 of Spades.

b

The first card is Diamonds and the second card is a 10.

c

The first card is a Queen and the second card is black.

d

The first card is not a 7 and the second card is not Spades.

18

Valentina randomly selects three cards, with replacement, from a normal deck of cards. Find the probability that:

a

The cards are five of clubs, King of clubs, and Jack of spades, in that order.

b

The cards are all red.

c

The first card is a 2, the second card is a spade and the third card is red.

d

The cards are all spade.

e

None of the cards is a 8.

19

In a lottery there are 37 balls.

a

Find the probability of a particular ball being drawn first.

b

A ball is discarded after it has been drawn. If ball number 22 is drawn on the first go, find the probability of ball number 15 being drawn next.

c

Is the probability of each successive ball drawn the same as the probability of the first ball drawn?

d

Determine whether the draws are dependent or independent events.

20

Two cards are to be chosen from a pack of cards numbered 1 to 11. Find the probability of drawing a 1, 2 or 3 first, and after replacing the card, drawing a 9, 10 or 11.

21

Three marbles are randomly drawn with replacement from a bag containing 6 red, 5 yellow, 6 white, 1 black and 3 green marbles. Find the probability of drawing:

a

Three white marbles.

b

No green marbles.

c

At least 1 red marble.

d

At least 1 white marble.

22

There are 4 blue counters and 6 brown counters in a bag. Find the probability of choosing a blue counter, not replacing it, then choosing a brown counter.

23

A number game uses a basket with 10 balls, all labelled with numbers from 1 to 10. Two balls are drawn at random.

Find the probability that the ball labelled 2 is picked once if the balls are drawn:

a

With replacement.

b

Without replacement.

24

A standard deck of cards is used and 3 cards are drawn out.

Find the probability that all 3 cards are clubs if the cards are drawn:

a

With replacement.

b

Without replacement.

25

A hand contains a 10, a jack, a queen, a king and an ace. Two cards are drawn from the hand at random, in succession and without replacement. Find the probability that:

a

The ace is drawn.

b

The king is not drawn.

c

The queen is the second card drawn.

26

A number game uses a basket with 5 balls, all labelled with numbers from 1 to 5. Three balls are drawn at random, without replacement.

a

Find the probability that the ball labelled 4 is picked.

b

Find the probability that the ball labelled 4 is picked and the ball labelled 1 is also picked.

27

From a set of 10 cards numbered 1 to 10, two cards are drawn at random without replacement. Find the probability that:

a

Both numbers are even.

b

One is even and the other is odd.

c

The sum of the numbers is 12.

28

Consider the word WOLLONGONG. If three letters are randomly selected from it without replacement, find the probability that:

a

The letters are W, O, L, in that order.

b

The letters are O, N, G, in that order.

c

All three letters are O.

d

None of the three letters is an O.

29

Three marbles are randomly drawn without replacement from a bag containing 6 red, 6 yellow, 6 white, 6 black and 4 green marbles. Find the probability of drawing:

a

Three white marbles.

b

Three black marbles.

c

Zero green marbles.

d

Zero yellow marbles.

e

At least one red marble.

30

Eileen randomly selects two cards, with replacement, from a normal deck of cards. Find the probability that:

a

The first card is a queen of spades and the second card is a 4 of clubs.

b

The first card is spades and the second card is a 4.

c

The first card is a Queen and the second card is black.

d

The first card is not a 7 and the second card is not Clubs.

31

Sarah has a box of Favourites chocolates. In this box there are 30 chocolates, 5 of which are Turkish Delights. Sarah takes and eats a chocolate without looking until she gets a Turkish Delight.

a

Find the probability she only eats one chocolate.

b

Find the probability she eats only two chocolates to the nearest percent.

c

Find the probability she eats five chocolates to the nearest percent.

d

As she eats more and more chocolates, state whether the probability of the next chocolate being a Turkish Delight is getting higher or lower.

e

How many chocolates must Sarah have eaten to be certain that the next chocolate will be a Turkish Delight?

32

Tom has shuffled a standard pack of 52 playing cards. He draws one card from the pack at a time, and then puts it aside. He will stop when he draws the 9 of Diamonds.

a

Find the probability he only has to draw exactly 1 card.

b

Find the probability he has to draw exactly 2 cards.

c

Find the probability he has to draw exactly 3 cards.

d

Find the probability Tom will stop after the 44th card.

e

As more cards are drawn that aren't the desired card, does the probability that the next card selected is the 9 of Diamonds increase or decrease?

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Outcomes

VCMSP347

Describe the results of two- and three-step chance experiments, both with and without replacements, assign probabilities to outcomes and determine probabilities of events. Investigate the concept of independence.

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