Mathematics Year 6: Equal chances, equal probabilities
Mathematics, Year 6 (Portugal): Not every random situation has equally possible results; when it does, their probabilities are equal, and an experiment helps to decide.
Equal chances
In a random situation you cannot know the result before it happens. Sometimes it is reasonable to assume that all results have equal chances of coming up: a balanced coin, a balanced die, identical slips of paper well mixed in a hat. In that case their probabilities are equal. In other situations it is not so: a thumbtack tossed in the air, tomorrow's weather, the result of a match. To decide, ask whether there is any reason for one result to be favoured.
Why symmetry counts
A balanced coin favours neither face: they are alike and the toss is random, so we assume equal chances. Whoever tosses changes nothing: the probability does not depend on whether Rui or Rita rolls the die. A thumbtack is different: the point and the head are not symmetrical. When you do not know, you experiment: repeat many times and see whether the relative frequencies come out close. That is how you decide whether a game is fair.
A coin and a thumbtack
You tossed a coin 100 times: 52 heads and 48 tails. Relative frequencies: 52/100 = 52% and 48/100 = 48%, very close. That supports equal chances: the probabilities of heads and tails are equal, even though the counts are not. Then you tossed a thumbtack 100 times: 63 with the point up and 37 with the point down, that is, 63% and 37%. A gap of 26 points is large: it is not reasonable to assume equal chances.
Two results are not “fifty-fifty”
Many people think: there are two results, so each has 50%. But “equal chances” only holds when there is no reason to prefer either. On a die, “rolling a 6” and “not rolling a 6” are two results, but “rolling a 6” is one face and “not rolling a 6” is five: they do not have equal chances. Winning or losing a game, or rain versus no rain, are not equally possible just because there are two. Before saying “equal”, ask: what makes them equally possible?
Fair games and draws
Tossing a coin to decide who starts works because heads and tails have equal chances: nobody is favoured. The same goes for the die of a board game or for a draw of names, if the slips are identical and well mixed. Whoever designs a game asks whether the results have equal chances; whoever plays can test it with an experiment. If a spinner has unequal sectors, it is no longer fair.
Test what you learned
On a balanced die, the faces 2 and 5 have equal chances of coming up. What can you say about their probabilities?
- a) The probability of 5 is greater, because 5 is greater than 2
- b) The probability of 2 is greater, because small numbers come up more
- c) They are equal
- d) They cannot be compared until the die is rolled many times
Correct answer: c) They are equal — Yes: if two results have equal chances of occurring, their probabilities are equal.
A thumbtack was tossed 100 times: 65 with the point up and 35 with the point down. Which conclusion is more reasonable?
- a) Equal chances, because there are two results
- b) Equal chances, because 65 + 35 = 100
- c) Nothing can be said about a thumbtack
- d) It is not reasonable to assume equal chances: 65% and 35% are far from 50% each
Correct answer: d) It is not reasonable to assume equal chances: 65% and 35% are far from 50% each — Right: 65/100 = 65% and 35/100 = 35%, a gap of 30 points, so point up seems clearly favoured.
A balanced die is rolled. “Rolling a 6” and “not rolling a 6” are two results. Do they have equal chances?
- a) No: “6” is one face and “not 6” is five faces
- b) Yes: there are two results, so 50% each
- c) Yes: a balanced die gives equal chances to everything
- d) It depends on who rolls the die
Correct answer: a) No: “6” is one face and “not 6” is five faces — Right: the six faces are equally possible, but the two results are made of different numbers of faces.
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