Probability

A spinner is used $30$ times. It lands on red $18$ times and on blue $12$ times. Complete a frequency table for the two outcomes and work out the relative frequency of red.

A frequency tree describes $80$ visits to a kiosk. On $52$ visits a hot drink is chosen and on the remaining visits a cold drink is chosen. A snack is also bought on $31$ hot-drink visits and on $7$ cold-drink visits. Complete all terminal frequencies and work out the percentage of visits on which a snack is bought.

Two weeks of trials are combined. In week 1, $72$ of $120$ trials succeed; $45$ successful trials and $18$ unsuccessful trials are fast. In week 2, $99$ of $180$ trials succeed; $66$ successful trials and $27$ unsuccessful trials are fast. Construct a combined frequency table for success or failure against fast or not fast. Compare the proportions that are fast in the two outcome groups.

A fair six-sided die is rolled $240$ times. Work out the expected number of rolls on which the score is greater than $4$.

A fair spinner has four equal sectors labelled $1$, $1$, $2$ and $3$. It is spun $600$ times. Work out the expected total of all the scores.

A box contains $3$ green, $2$ yellow and $1$ red token. A token is chosen at random and replaced. A player receives $\pounds4$ for green, $\pounds1$ for yellow and nothing for red. The entry fee is $\pounds2.20$. Work out the organiser's expected profit from $900$ plays and decide whether the game is fair to the player.

Place these events in order from least likely to most likely: event A has probability $0.72$, event B has probability $\dfrac{1}{4}$, and event C has probability $0.5$.

An outcome occurs $84$ times in $140$ trials. Its theoretical probability is $\dfrac{5}{8}$. Work out the relative frequency and the expected frequency in $560$ further trials. Comment on the experimental result.

A model says that a device flashes with probability $0.5$ on each trial. In a short run it flashes $41$ times in $80$ trials. After $500$ trials in total it has flashed $247$ times. Calculate both relative frequencies and decide which result gives stronger evidence about the model.

The probability that a parcel arrives early is $0.37$. Work out the probability that it does not arrive early.

A trial has four mutually exclusive outcomes A, B, C and D. Their probabilities are $0.28$, $0.41$, $x$ and $0.19$ respectively. Work out $x$.

For two events A and B, $P(A)=0.58$, $P(B)=0.47$ and $P(A\cap B)=0.21$. Work out the probability that neither A nor B occurs.

A fair coin gives a relative frequency of heads of $0.64$ after $25$ tosses. Explain what is likely to happen to the relative frequency as many more unbiased tosses are made.

A spinner is designed to land on green with probability $0.3$. It lands on green $8$ times in $20$ spins and $63$ times in $200$ spins. Compare the two empirical probabilities with the theoretical probability.

A fair die is tested. In the first $60$ rolls, a six appears $16$ times. After $600$ rolls in total, a six has appeared $108$ times. A student says the die must be biased because neither relative frequency equals $\dfrac{1}{6}$. Evaluate the student's claim.

A uniform is made from one of two shirts, blue or white, and one of three ties, red, silver or green. List all possible shirt-and-tie combinations.

The universal set is the integers from $1$ to $20$. Set A contains the multiples of $3$ and set B contains the factors of $18$. Enumerate the four regions of a Venn diagram for A and B.

A three-digit number is formed from three different digits chosen from $1$, $2$, $3$ and $4$. Enumerate all the numbers that are greater than $230$ and even. How many of these numbers do not contain the digit $1$?

A fair coin is tossed and a fair spinner labelled $1$, $2$, $3$ is spun. Write the six outcomes as ordered pairs and find the probability of getting a head and an even number.

Spinner A has equal sectors labelled $1$, $2$, $3$. Spinner B has equal sectors labelled $1$, $2$, $3$, $4$. Construct a possibility space and find the probability that the two scores have a sum greater than $5$.

Two different cards are chosen in order from cards labelled $1$, $2$, $3$, $4$ and $6$. The first card is the tens digit and the second is the units digit. Construct the theoretical possibility space and find the probability that the two-digit number is divisible by $3$.

A fair coin is tossed and an independent spinner lands on red with probability $\dfrac{3}{5}$. Work out the probability of a head and red.

A bag contains $4$ blue counters and $3$ amber counters. Two counters are chosen at random without replacement. Work out the probability that both counters are blue.

A bag contains $5$ black and $3$ white counters. Three counters are chosen at random without replacement. Work out the probability that exactly two are black. State why the branch probabilities change after each choice.

A two-way table records $60$ students. Of the $24$ who travel by bus, $9$ arrive late. Find the probability that a randomly chosen bus traveller arrives late.

Of $500$ expected customers, $60\%$ are members. Of the members, $70\%$ order online. Of the non-members, $35\%$ order online. Use expected frequencies to find the probability that an online customer is a member.

In an expected-frequency Venn diagram for $240$ people, $138$ are in set A, $102$ are in set B and $54$ are in both sets. A person is chosen from those who are in exactly one of the sets. Find the conditional probability that the person is in A.