Developing an Intuitive Understanding of Probability

Anela throws a standard six-sided die (marked from 1 to 6) while Zeke throws a standard eight-sided die (marked from 1 to 8).

The probability Zeke throws a larger number than Anela is...

• more than 50%

• exactly 50%

• less than 50%

Answer explanation

The grid shows every possible combination, and the green-marked squares show places where Zeke rolled a larger number than Anela. The green squares cover more than half of the grid, so the probability Zeke will roll larger is more than 50%. (The exact amount is 27 out of 48, or 56.25%.)

You throw a pair of fair coins. The probability that the result is either both are heads or both are tails is ...

• more than 50%

• exactly 50%

• less than 50%

Answer explanation

The four possible coin flips are

heads-heads

heads-tails

tails-heads

tails-tails

and all four are equally likely. Two have our goal outcome (either both heads or both tails), so our probability of reaching it is exactly 50%.

Where Cormac lives, it is sunny three-quarters of the days and rains for the rest.

Every day Cormac makes a weather prediction by flipping a fair coin; heads mean predicting sun and tails mean predicting rain.

The probability Cormac will correctly predict the weather this way on a particular day is

• more than 50%

• exactly 50%

• less than 50%

Answer explanation

There are 8 outcomes when charted this way, and in exactly half of them the prediction is correct.

A standard deck with 52 cards — 26 red and 26 black — is shuffled.

The probability that the top card is the same color as the bottom card is

• more than 50%

• exactly 50%

• less than 50%

Answer explanation

Suppose the top card is red. Then of the 51 cards remaining, 25 are red, so there is roughly

49% chance the bottom card is also red.

Suppose the top card is black. Then of the 51 cards remaining, 25 are black, so there is

49% chance the bottom card is also black.

Since red and black might be the top card with equal probability, the overall probability of the top and bottom colors matching is roughly

49%.

Let's modify the problem you saw earlier with Anela and Zeke slightly:

Zeke still throws a standard eight-sided die (marked from 1 to 8). However, Anela now throws a non-standard six-sided die that is marked from 2 to 7.

The probability Zeke throws a larger number than Anela is

• more than 50%

• exactly 50%

• less than 50%

Answer explanation

The green squares show where Zeke rolls larger than Anela; this is 21 out of 48 squares or 43.75%.