# INVESTIGATION Beat the dealer

In document NELSON SENIOR MATHS ESSENTIALS 12 (Page 83-86)

To complete this group activity, you will need a pack of normal playing cards, a pen and a record sheet. You can use the 'Beat the dealer' record sheet from NelsonNet.

Rules of the game

• The dealer shuffles the cards, then deals one card face up to each player. The dealer deals one card to themself face down. Then the dealer deals another card face up to each player and a card face down to themself.

• In this game, the cards have point values that are equal to the number showing on the cards.

All jacks, queens and kings score 10, and aces score 11. The jokers are not used.

• Players calculate the total value of their pair of cards.

• The dealer reveals the face-down cards and calculates their total value.

• A player wins if their total is higher than the dealer's total. If the dealer's total is equal to or higher than the player's total, the dealer wins.

What you have to do

In your group, nominate a dealer, a recorder and several players.

2 Play the game 10 times. Each time, record who wins out of the dealer and each player.

3 Count the number of times the players won.

4 Calculate the total number of games played, that is, 10 x the number of players.

5 Determine the experimental probability of beating the dealer by dividing your answer to question 3, the number of times the players beat the dealer, by your answer to question 4, the number of games.

Class discussion questions

Who is more likely to win this game: the players or the dealer? Why?

2 If you play this game in a casino, every game costs \$1. You receive \$2 if you beat the dealer and nothing if you don't. If you play 20 games in a casino, are you likely to win or lose money?

### 0 Example 1

Ruth catches the train to school. She recorded that on 40 consecutive trips, she got a seat only 16 times. What is the experimental probability that Ruth will get a seat on the train when she is going to school?

### Solution

Ruth got a seat 16 out of 40 times. The experimental probability will be a fraction with 16 on the top and 40 on the bottom. You can use the fraction key on your calculator to simplify the fraction.

Experimental probability= :�

_2

5

Approximately 2 days out of every 5, Ruth will

### 0 Example 2

Richard recorded the direction of the wind and whether rain followed. He found that, out of 30 times when the wind came from the north-west, rained followed 20 times. What is the experimental probability that in Richard's local area it will rain after wind blows in from the north-west?

### Solution

It rained 20 times out of 30. The experimental probability is a fraction with 20 on the top and 30 on the bottom.

Experimental probability= 20

30 _2 -3

### Experimental probability

To complete the questions in this exercise, you will need a pack of 52 playing cards, two coins and a normal six-sided die.

4%,j,,i,jiiti ;;i Toss one coin 20 times and record the number of heads.

b What is the experimental probability of obtaining a head when you toss a coin?

c Will you always get the same value for the experimental probability of tossing a head? Why, or why not?

2

### ;;.;,,;.,;a

Roll a normal six-sided die 30 times and record each number that shows. Use your results to determine the answers to the following questions.

a What is the experimental probability of rolling a 6 on a die?

b Determine the experimental probability of obtaining a number less than 3 when you roll a normal die.

The expression 'less than 3' does NOT include 3. The numbers on a die that are less than 3 are 1 and 2.

3 When you toss a pair of coins together, you can get two heads, two tails or a head and a tail. a Toss a pair of coins together about 30 times and record the numbers of times you get two

b Determine the experimental probability of obtaining two heads when you toss a pair of coins. c Which is more likely to happen when you toss a pair of coins: obtaining two tails or

obtaining a head and a tail?

d When you toss a pair of coins, are you more likely to get two things the same or two different things?

4 A normal pack of playing cards contains 52 cards that are arranged in four suits. The suits are hearts •, diamonds ♦, spades it. and clubs of.. Assign each card the same number value as in the game 'Beat the dealer' on page 72. Shuffle the cards and deal pairs of cards.

a What is the experimental probability that the sum of the values of a pair of cards will be 14? b Determine the experimental probability that the sum of the values of a pair of cards will be

bigger than 15.

c What is the experimental probability that a pair of cards will be the same suit?

5 Liam invented his own rules for a game he calls 'In the space'. Liam's game is played with a normal pack of playing cards. He deals a pair of cards to each player. If the two cards are consecutive numbers or the same number, he deals a third card to replace one of the first two cards. Then he deals each of the players an additional card. The players win if their additional card has a value between the values of the other two cards.

a Play Liam's game in your group and record the number of times the players win and the number of times they lose.

b What is the experimental probability of a particular player winning this game?

c Liam charges people \$1 to play the game and he gives them \$3 if they win. Do you think the game is fair? Why, or why not?

### 1111 THEORETICAL PROBABILITY

In the previous section, you used experiments to determine probabilities. In this section you will learn about calculating theoretical probabilities. Maya is hoping to score a 5 when she rolls a normal six-sided die. When she rolls the die, the number showing on the top of the die could be l, 2, 3, 4, 5 or 6, and each number is just as likely to show as any of the others. There is only one 5 from a possible six numbers. The probability or chance that she will roll a 5 is one number out of

. b 1

SlX num ers, or 6.

The probability that an event E will happen is:

P(E) = number of ways of obtaining the desired result

total number of equally likely res1;1lts

In document NELSON SENIOR MATHS ESSENTIALS 12 (Page 83-86)

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