Probability

35

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

为什么Written 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

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

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

1/6

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

为什么Every 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

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

为什么A 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

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

为什么Writing 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

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

为什么All 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

提示Every result on the first dice can pair with every result on the second.

为什么A 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

提示Multiply the chance of the event by the number of attempts.

为什么Expected 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

提示Compare the number of successes with the number of tries.

为什么Dividing 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

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

为什么Each 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

提示If one of them occurs, the other is ruled out.

为什么Rolling 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

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

为什么Independent 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.