4.5 Concepts of probability questions for IB Maths AI

Write down the number of outcomes in the sample space.

Find the probability that her score is 12.

Find the probability that her score is an odd number.

Hence write down the probability that her score is an even number.

Find the number of edges of $G$.

Explain why $G$ cannot be a tree.

Two of the eight campers are chosen at random. Find the probability that they are friends.

A ninth camper, Pavla, joins the camp and makes friends with $k$ of the other campers. No other friendships change.

Find the least value of $k$ for which the probability that two campers chosen at random from the nine are friends is greater than 0.6.

Consider again the original eight campers. Show that $G$ must be connected.

Write down the relative frequency of Kofi scoring from a penalty kick.

Use your answer to part (a) to estimate the probability that Kofi does not score from his next penalty kick.

Next season Kofi expects to take 24 penalty kicks. Estimate the number of these from which he will score.

State one reason why your estimate in part (c) may not be reliable.

Find $\mathrm{P}(S=4)$.

Complete the following table to show the probability distribution of $S$.

Find $\mathrm{E}(S)$.

Players pay 1 US dollar (USD) to play. A player who scores 7 receives 5 USD and a player who scores 5 or 6 receives 1 USD. Otherwise the player receives nothing.

Find the expected gain of a player per game. Give your answer as a fraction.

Hamid suspects that spinner $B$ is not fair. He plays the game 120 times and records the following scores.

He performs a ${\chi }^2$ goodness of fit test at the 5% significance level, with the hypotheses

H₀: the scores follow the distribution found in part (b)

H₁: the scores do not follow the distribution found in part (b).

(i) Write down the expected frequencies.

(ii) Write down the number of degrees of freedom.

(iii) Find the $p$-value for the test.

(iv) State the conclusion of the test. Give a reason for your answer.

List the sample space for this game.

Find the probability that a player obtains a head and the spinner does not land on blue.

A player wins a prize when they obtain a tail and the spinner lands on red. During the fair, the game is played 150 times.

Find the expected number of prizes won.

Write down the relative frequency of a score of 1.

Find the number of times that each score would be expected to occur in 300 rolls if the die were fair.

Ama performs a ${\chi }^2$ goodness of fit test at the 5% significance level to determine whether the die is fair.

Write down the null hypothesis for this test.

Write down the number of degrees of freedom.

Find the $p$-value for the test.

State the conclusion of the test. Give a reason for your answer.

Ama says that the probability of rolling a 1 with this die should be taken as your answer to part (a). Comment on her statement.

In one of the company's games, a player wins a token if they roll a 1 at least once in three rolls of the die. Assume that the die is fair.

(i) Find the probability that a player wins a token.

(ii) The game is played 80 times. Find the expected number of tokens won.