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7.06 Applications of probability

Worksheet
Probability in games
1

Find the probability of picking the ten of diamonds from a standard deck of cards.

2

A local raffle prize will be given to the person who has the winning number from one of the 200 tickets sold. Find the probability of winning the raffle if you purchase 30 tickets.

3

In a card game, Aaron is dealt a hand of two cards from a standard deck of 52 cards.

a

How many possibilities are there for the first card?

b

How many possibilities are there for the second card?

c

Hence, calculate the total number of ways the cards could be dealt.

d

If Aaron is dealt the King of clubs and the three of hearts, is this the same as being dealt the three of hearts and the King of clubs?

e

Hence determine the number of different hands he could be dealt.

f

State the probability his hand consists of the two of spades and the four of clubs.

4

In Monopoly, each player rolls two dice on their turn. The sum of the two dice tells them how many squares they move. It is Beth's turn and she is 5 squares away from “Free Parking”, where she wants to land. What is the probability that she will land on “Free Parking” on this turn?

5

Christa enters a competition in which she guesses the 3-digit code (from 000 to 999) which cracks open a vault containing one million dollars. If the 3-digit number to open the vault is randomly generated by a computer, find the probability that it is:

a

The number 123.

b

An odd number.

c

An even number (including 000).

d

A number greater than 123.

e

A number divisible by 10.

f

A number less than 321.

6

In a game of Blackjack, a player is dealt a hand of two cards from the same standard deck. Find the probability that the hand dealt:

a

Is a Blackjack (an Ace paired with 10, Jack, Queen or King).

b

Has a value of 20 (Jack, Queen and King are all worth 10 and an Ace is worth 1 or 11).

7

In a game of Draw Poker, a player is dealt a hand of 5 cards from the same deck.

a

How many possible hands are there?

b

Find the probability of being dealt a:

i

Flush (five cards of the same suit).

ii

Royal flush (10, Jack, Queen, King and Ace of the same suit).

8

In a game of Monopoly, rolling a double means rolling the same number on both dice. When you roll a double this allows you to have another turn. Find the probability that Sarah rolls:

a

A double 1.

b

A double 5.

c

Any double.

d

Two doubles in a row.

9

In a game of Yahtzee, each player rolls 5 dice. A player rolls two of the dice and gets 2 sixes. The player still needs to roll the other 3 dice. Find the probability that they will get:

a

Yahtzee (5 of the same number).

b

4 of a kind.

c

3 of a kind.

10

In a game of Draw Poker, a player is dealt a hand of 5 cards where each card is dealt from a brand new deck. Find the probability of being dealt a:

a

Flush (five cards of the same suit).

b

Royal flush (10, Jack, Queen, King and Ace all from the same suit)

11

One of the better hands in poker is a "two pair", where a pair is two cards that have the same number or value. For example, two threes will form a pair. I draw four cards from a standard deck, and they consist of one pair and two other cards that don't match with any card in my hand.

If I draw another card, what is the probability that this card will give me a second pair?

12

One of the better hands in poker is a "three of a kind", where cards are the same number or value. For example, three fours makes three of a kind. I am dealt four cards from a standard deck consisting of one pair and two other cards that do not match any of my other cards.

If I draw another card from the remaining deck, what is the probability that this card will give me three of a kind?

13

One of the better hands in poker is a flush, where all five cards are of the same suit. If I have three spades already and am about to draw two cards, what is the probability that I will get a flush?

14

A jar contains 7 blue M&M’s and 5 red M&M’s. Bob takes one from the jar and does not replace it. Luke then takes an M&M from the jar. If it is the same colour as Bob’s, Luke wins the jar of M&M's.

Find the probability that Luke wins the jar.

Traffic light problems
15

A traffic light pole with green, red and yellow lights is set up on a race track. A race car approaching the pole is equally likely to see a green, red or yellow light.

Select the best reason for this:

A

The three lights stay on for the same amount of time.

B

The race car driver is driving at a constant speed.

C

The lights are the same size.

16

A set of traffic lights is red for half the time, orange for \dfrac{1}{5} of the time and green for the rest of the time. Find the probablity that the traffic light is green.

17

A traffic light is green for 35 seconds, yellow for 5 seconds and then red for 50 seconds. Find the probability that the next person at the traffic lights gets a:

a

Red light.

b

Yellow light.

c

Green light.

18

At a particular traffic light, it was found that the light in a particular direction stays green for 119 seconds, yellow for 5 seconds and red for 76 seconds.

If a car approaches the traffic light, find the probability, as a percentage, that the light will be:

a

Green

b

Yellow

c

Red

19

At a railway crossing, there is only a red and green light, and they stay on for the same amount of time.

a

If you cross the railway twice a day, list the outcomes for the possible colours of the lights.

b

Find the probability you get a green light once, and a red light once in the day.

20

On the island of Timbuktoo, the probability that a set of traffic lights shows red, yellow or green is equally likely. Christa is travelling down a road where there are two sets of traffic lights.

Consider the tree diagram that indicates the possible pairs of traffic light signals:

a

Find the probability that both sets of traffic lights will be yellow.

b

Find the probability that Christa will have to stop at least once.

c

Find the probability that Christa will not have to stop at all.

21

A traffic light at a railway crossing has a chance of 0.3 of being yellow. Three cars approach the light separately at different times throughout the day.

Find the probability that all three cars approach a yellow light.

22

At an intersection, the traffic lights for different directions each have a probability of 0.35 of being green. Two cars travelling in different directions arrive at the traffic lights at the same time.

Find the probability that:

a

They both get a green light.

b

Exactly one of them gets a green light.

c

They both get a red or yellow light.

23

Every 60 seconds, a traffic light remains green for 34 seconds, yellow for 3 seconds and red for 23 seconds.

a

Which outcome is more likely: arriving at the traffic light when it is yellow, or arriving at the traffic light when it is green?

b

On her way to work, Nadia passes through 3 sets of such lights. What is the probability that none of the lights are green?

c

Find the probability that the first two sets of lights are green and that the third set is red.

24

Tina is at point A driving in the direction of point B. There are traffic lights at points B, C, D and E, and the probability of a green light at any one of these is 0.4.

If Tina arrives at a traffic light when it is green, she will go straight. If it is not green, she will turn right. She does not turn left or drive back in the direction of A at any time.

a

Find the probability that Tina drives to point P.

b

Find the probability that Tina drives to point Q.

25

The data below shows the time spent waiting for a green light at a set of traffic lights, in seconds:

11,\, 119,\, 5,\, 74,\, 32,\, 90,\,31,\,66,\,91,\,33,\,81,\,37,\,94,\,17,\,84,\,101,\,56,\,41,\,14,\,52
a

Complete the following relative frequency table:

TimeFrequencyRelative Frequency
0\leq t\lt 101
10\leq t\lt 20
20\leq t\lt 30
30\leq t\lt 40
40\leq t\lt 50
50\leq t\lt 602
60\leq t\lt 70
70\leq t\lt 801
80\leq t\lt 902
90\leq t\lt 100
100\leq t\lt 110
110\leq t\lt 120
b

Construct a relative frequency histogram.

c

Calculate the probability of a person waiting being between 20 and 29 seconds, inclusive.

d

Calculate the probability of a person waiting less than 30 seconds.

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Outcomes

ACMEM157

determine the probabilities associated with simple games

ACMEM158

determine the probabilities of occurrence of simple traffic-light problems

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