Practice: Probability · The Lumenary

Probability

Warm-upHuman

A coin is tossed and a die is rolled. How many outcomes are in the sample space? (For example, (H, 3) is one outcome.)

Probability

Warm-upHuman

Two fair dice are rolled. What is the probability that the sum is $8$ ?

Choose one

\tfrac{5}{36}

\tfrac{1}{11}

\tfrac{3}{36}

\tfrac{8}{36}

Probability

CoreHuman

A die is rolled three times. What is the probability of getting at least one six?

Choose one

1 - \left(\tfrac56\right)^3 = \tfrac{91}{216}

\tfrac16 + \tfrac16 + \tfrac16 = \tfrac12

\left(\tfrac16\right)^3 = \tfrac1{216}

\tfrac{3}{18}

Probability

CoreHuman

A number is chosen at random from $1, 2, \ldots, 20$ (all equally likely). What is the probability that it is a multiple of $2$ or a multiple of $5$ ?

Choose one

\tfrac{12}{20}

\tfrac{14}{20}

\tfrac{2}{20}

\tfrac{10}{20}

Probability

Warm-upHuman

A fair coin has landed tails $6$ times in a row. What is the probability that the next toss is heads?

Choose one

\tfrac12

More than

\tfrac12

, because heads is dueLess than

\tfrac12

, because tails is on a streak

\left(\tfrac12\right)^7

Probability

CoreHuman

In 1754 d'Alembert argued: “Toss a coin up to twice, stopping when heads appears. The possible results are H, TH and TT; two of the three give a head, so the probability of getting a head is $\tfrac23$ .” What is wrong?

Choose one

The three results are not equally likely: H has probability

\tfrac12

, TH and TT

\tfrac14

eachNothing — the answer

\tfrac23

is rightThere should be only two results, H and THe should have multiplied instead of counting

Probability

CoreHuman

Neema rolls a fair die $60$ times and gets $14$ sixes. Which statements are true? (Choose all that apply.)

Choose all that apply

The relative frequency of a six was

\tfrac{14}{60}

The die must be unfair, because she should have got exactly 10 sixesThe probability of a six is still

\tfrac16

for a fair dieWith 6,000 rolls, the relative frequency would very probably be closer to 1/6

Probability

StretchHuman

Three dice are rolled. In how many of the $216$ ordered outcomes is the sum equal to $10$ ?

Probability

CoreFormal

In the Lean file, prob S A is the probability of the event A in the list of outcomes S. Which statement is the complement rule, correctly stated?

Choose one

S ≠ [] → prob S (fun x => !A x) = 1 - prob S Aprob S (fun x => !A x) = 1 - prob S AS ≠ [] → prob S (fun x => !A x) = prob S A - 1S ≠ [] → prob S A = 1

Probability

StretchComputational

Write prob_sum(dice, total) that returns the exact probability (as a Fraction) that dice fair dice add up to total. List every ordered outcome with itertools.product(range(1, 7), repeat=dice) and count. Then check Galileo: is a total of $10$ more likely than $9$ with three dice?

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