Probability

35

HintRewrite all three in one common form before deciding which sits highest on the scale.

The whyWritten as percentages the three are 35%, 25% and 30%, so 0.35 is the most likely. Fractions, decimals and percentages all mark positions on the same 0 to 1 scale, and converting to one form is the only fair way to rank them.

Evens

HintIt is just as likely to happen as not, like tossing a fair coin.

The whyAn evens chance means the two possibilities are equally likely. The chance words run impossible, unlikely, evens, likely and certain.

1/6

HintCount the faces that win, then compare that with how many faces there are altogether.

The whyEvery face is equally likely, so the probability is successful outcomes over total outcomes. Written as a decimal it is about 0.17.

It is biased towards six

HintWork out how many you would expect from a fair one, then judge whether the gap is too big to be chance.

The whyA fair dice would give about 600 ÷ 6 = 100 sixes. Getting 180 in 600 rolls is far more than random wobble over so many trials, so the experimental probability of 0.3 points to bias. The same gap in only 6 rolls would prove nothing.

So none are missed or repeated

HintWorking at random makes it easy to lose your place, and a fixed order does not.

The whyWriting HH, HT, TH, TT in a fixed order guarantees the set is complete. A complete set is what makes any probability counted from it trustworthy.

0.7

HintEither it happens or it does not, and those two possibilities cover everything.

The whyAll the outcomes together have a probability of 1, so the chance of an event not happening is 1 minus the chance that it does. This is often the quickest route to an answer.

36

HintEvery result on the first dice can pair with every result on the second.

The whyA 6 by 6 sample space grid holds one cell for each pair, making it a complete map of the possibilities. A question such as the chance of a total of 7 can then be read straight off it.

10

HintMultiply the chance of the event by the number of attempts.

The whyExpected frequency is probability × trials, so a sixth of 60 gives the estimate. It predicts the long run, not the result of any one set of rolls.

0.15

HintCompare the number of successes with the number of tries.

The whyDividing 30 by 200 gives the estimate. Experimental probability is the only route when outcomes are not equally likely, and it grows more reliable as the number of trials rises.

Random variation

HintReal results wobble around what the theory predicts, especially over a short run.

The whyEach toss is independent, so the count drifts either side of the expected value. The more tosses are made, the closer the proportion usually gets to a half.

Mutually exclusive

HintIf one of them occurs, the other is ruled out.

The whyRolling a 2 and rolling a 5 on one dice are mutually exclusive, so those probabilities may be added. Events that overlap, such as even and more than 3, cannot simply be added.

1/2

HintThe coin has no memory of what it did before, so nothing has changed.

The whyIndependent events do not influence one another, so the earlier run tells you nothing about the next toss. Believing that a tail is now due is known as the gambler's fallacy.