In general, people don't have a good understanding of chance. People make bad decisions involving

. chance and casinos make money from gambling fallacies. While calculating probabilities can be

tricky, there are two things we should remember when trying to decide whether a chance statement is correct.

ur _{1 } _{h c } _{1 P( } _{) the total number of ways the event can happen h h }
• vve can on y use t e 1ormu a event

### =

_{th t t 1 }

_{e o a num er poss1 111es }

_{b }

_{·b·l·t· }w en t e

total number of possibilities are equally likely.

• In a fair game of chance, for example, coin-tossing or roulette, the equipment does NOT remember what has happened in the past. The probability at each occurrence remains the same.

### 0 Example 14

Decide whether this statement is fact or fallacy.

Scott is going to swim from Newcastle to Sydney. There are two possibilities: either he will swim the distance or he won't. Therefore, the probability that Scott will swim the distance is½·

### Solution

I h. n t 1s context, t e 1ormu a event -h c 1 P( ) _ the total number of ways the event can happen _{t e tota num er poss1 1tles }_{h } _{1 } _{b } _{"bil" . } cant e, b

used. The argument is incorrect because the 2 possibilities, 'swimming the distance' and 'not swimming the distance' are NOT equally likely.

### 0 Example 15

A group of people are betting on the result of a coin toss. On the last 6 tosses, the coin has landed showing heads. Gavin said, Tm going to bet on tails. The coin is more likely to land showing tails next time because a run of 7 heads is extremely unlikely:

Is Gavin's thinking correct?

**Solution **

Gavin is correct that a run of 7 heads is unlikely. But in this case 6 heads have happened already. The coin doesn't remember what has happened in the past. Irrespective of the previous

outcomes, the probability that the next toss will result in a tail is ½· Gavin's statement that the next result is more likely to be a tail is incorrect.

**EXERCISE 4.13 **

**Fact or fallacy? **

*Group discussion exercise. *

In your group, discuss each of the following probability statements and decide if they're facts or fallacies. Be ready to justify your group's opinion when other groups disagree!

**ii****,****i,,i****,****jJii **

I will either die on my birthday or I won't die on my birthday. That's 1 possibility out
of 2. The probability that I will die on my birthday is½·
2 When you toss 2 coins, there are 3 possible outcomes: 2 heads, 2 tails or a head and a tail. The probability of getting 2 heads when you toss a pair of coins is½·

3 There are 10 runners in a race.

4 Melissa has 4 sons and she would like to have a daughter. She correctly calculated that the probability of a family of 5 children all being boys is J__ She says that she is going to have

32

another baby because the probability that her next baby will be a girl is�, which is close to

. 32

a certainty.

5 1§,j_{,,}j,jjij Sharon is going to toss a coin 8 times and record the result.

a She is more likely to record T, T, H, T, H, H, T, H, T than H, H, H, H, H, H, H, H. b She is more likely to record 4 heads and 4 tails, than 2 heads and 6 tails.

6 There is a 60% chance of rain on Saturday and a 40% chance of rain on Sunday. There is a 100% chance of rain on the weekend.

7 There is a mixture of red and blue balls in a bag. The probability of selecting a red ball at random from the bag is .!..

2

8 Tony plays competition table tennis. He wins 3 out of 5 matches that he plays, making the probability that he will win any match ¾- He is playing in a 5 match competition and he has won the first 3 games. It is likely that he will lose the next 2 matches.

9 In NSW, lotto players select 6 numbers from the numbers 1 to 45.

a The numbers 3, 11, 15, 16, 25 and 31 are more likely to be the 6 winning numbers than the numbers 1, 2, 3, 4, 5 and 6.

b If you include some or all numbers bigger than 30 in your six numbers and your six numbers come up, you will win more money than if the 6 winning numbers included no number bigger than 30.

10 Kim saw a tall, attractive man with a beautiful girlfriend getting into a sports car. The man is more likely to be a professional tennis player than a primary school teacher.

11 Martin is interested in buying a house that is in a 1-in-a-hundred years flood zone. The house

flooded in March this year. It won't flood again for another 100 years.

12 Mandy needs an operation to remove a dangerous melanoma from her back. Her doctor said that she had a 98% chance of surviving the surgery. Mandy could die during the surgery .

**Probability of½ **

**Probability **
**crossword **