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 etotal 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