Two dice are rolled and the difference between the largest number and smallest number is calculated. A player wins \$1 if the difference is 3, \$2 if the difference is 4 , \$3 if the difference is 5 and \$0 otherwise.
Complete the following table, which shows the difference between the dice rolls:
Let X be the winnings from one game. Construct a probability distribution table for X.
Find the expected winnings.
If it costs \$2 to play each game, find the player's expected return.
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 0 | 1 | 2 | |||
| 2 | ||||||
| 3 | ||||||
| 4 | ||||||
| 5 | ||||||
| 6 |
X is a uniform discrete random variable that takes on the values 1, 2, 3 or 4.
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| P(X = x) |
Find E \left( X \right).
Two normal die are rolled and the sum of the numbers on the uppermost face recorded.
Let Y represent the value of the sum of the two die.
Complete the table for this discrete probability distribution.
| y | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| P(Y = y) | \dfrac{1}{36} | \dfrac{3}{36} | \dfrac{6}{36} | \dfrac{1}{36} |
State the most likely sum to occur.
Calculate the expected value of Y.
A game uses a spinner with numbers from 1 to 12, each outcome being equally likely. A player wins \$9 if the outcome is greater than 9, \$6 if the outcome is less than 5 and loses \$3 otherwise.
Let X be the return from one game. Complete the probability distribution table for X.
State the expected return in dollars.
| x | -\$3 | \$6 | \$9 |
|---|---|---|---|
| P(X = x) |
Xavier tosses two coins. He wins \$10 for two tails, \$5 for two heads, and nothing for a head and a tail.
Find the expected value of this game.
If the game costs \$5 to play, will Xavier likely to win or lose in the long run?
A deck of cards has cards numbered 8 through 16. A player draws a card at random. A player wins \$3 if the card is odd and loses \$3 if it's even.
Let X be the winnings of the player. Find the expected return in dollars.
An investment scheme advertises the following returns after 2 years based on historical probabilities:
Calculated the expected return on investment as a percentage.
If someone invested \$50\,000, how much could they expect their investment to be worth after 2 years?
| \text{Return }(x) | 10\% | 15\% | 25\% |
|---|---|---|---|
| P(X = x) | 0.8 | 0.1 | 0.1 |
For each of the probability distribution graphs below:
List the possible outcomes of X.
Is this a uniform or non-uniform distribution?
Find the expected value of X.
The Vaucluse Vigilantes and the Woollahra Weasels play a series of basketball matches and the first to win four games wins the series. The following distribution shows the probabilities of the total number of games that will be played in the series:
| X | 4 | 5 | 6 | 7 |
|---|---|---|---|---|
| p(X = x) | 0.11 | 0.34 | 0.33 | 0.22 |
Find the expected number of games that will be played in the series.
A salesperson is starting work in a new region and analyses the probability of how many sales he is likely to make in the next month:
| \text{Number of sales } (x) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| P(X = x) | 0.35 | 0.25 | 0.2 | 0.15 | 0.05 |
Given that he makes at least one sale, state the probability that he will make 2 sales.
The salesperson is offered 2 payment schemes:
-
Option A: Flat monthly income of \$1800
Option B: \$1000 flat fee per month plus \$500 per sale
If he chooses option B, what is his expected monthly income?
Which option should he choose to maximise his income?
A probability distribution function is represented in the table below.
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| P(X = x) | 3k^{2} | 2k | k | 4k^{2}+k | 2k |
Find the value of k.
Calculate P ( X \leq 4 )
Find P ( X > 1 | X < 4 )
Find E \left(X\right).
At a car park in the city, all day parking is charged on the following basis:
Cars with just a driver pay \$20
Cars with a driver and one passenger pay \$18
-
Cars with a driver and at least two passengers pay \$10
The number of people in one of these cars on a given day is summarised in the following table:
| Number of people | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Number of cars | 4 600 | 3 400 | 1 100 | 600 | 300 |
Calculate the probability a randomly selected car is carrying 3 people.
Given that a car was carrying at least 2 people, what is the probability it was carrying 4?
Let X represent the parking fee paid by a randomly selected car. Construct the probability distribution table for X.
Calculate the expected revenue per car in this car park.
Let X be a random variable with the following probability distribution table:
| x | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|
| P(X = x) | 0.2 | 0.6p^2 | 0.1 | 1 - p | 0.1 |
Find the value(s) of p.
Let p = \dfrac{2}{3}. Calculate:
When a student completes a task set by their teacher on Spacemaths, the number of hints used is monitored by the system.
The probability of using at least 1 hint is 0.6.
The probability of using 2 hints is the same as using 3 hints.
The probability of using 1 hint is the same as using 4 hints.
At most students can use 4 hints.
The probability they use 2 hints is half the probability that they use 0 hints.
Let X represent the number of hints they used. Complete the following probability distribution table:
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| P(X = x) |
Find the expected number of hints a student will use.
Given that a student used at least 2 hints, what is the probability they used 4 hints?
The table below represents a discrete probability distribution:
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| P(X = x) | 0.25 | m | 0.05 | n | 0.1 |
Use a property of probability distributions to express m in terms of n.
Use the fact that E \left(X\right) = 2.9 to express m in terms of n.
Hence solve for n.
Solve for m.
Xanthe is sitting a multiple choice quiz consisting of 3 questions, each with 4 possible answers. Xanthe hasn’t studied for the quiz so she will guess the answer to each question at random.
Construct a probability tree diagram to represent all possible combinations of which questions Xanthe got right and wrong.
Let X be the number of correctly answered questions. Construct a probability distribution table for X.
Determine the most likely number of questions Xanthe guessed correctly.
Calculate how many questions Xanthe was expected to answer correctly by guessing.
A cat has a litter of three kittens. Each kitten is equally likely to be born male or female.
Construct a probability tree diagram to represent all possible combinations of the sex of these three kittens.
Let M be the number of male kittens born to this cat. Construct the probability distribution table for M.
State the expected number of male kittens born.
Jo and Ky are playing a game of cards that is either won or lost, there is no draw. The probability that Jo wins the first game is 0.6.
-
If Jo wins a game, the probability he wins the next game is 0.7.
-
If Jo loses a game, the probability that Ky wins the next game is 0.8.
They keep playing until either Jo or Ky wins two games.
Construct a tree diagram of this situation.
Let X represent the number of games of cards played before someone wins two games. Construct a probability distribution table for X.
Find E \left(X\right).
Given that Jo won, calculate the probability that 3 games were played.
In a large batch of microphones, 10\% are known to be defective. An online retailer mails out 3 microphones, randomly selected from the batch.
Construct a probability tree diagram to represent all possible combinations of defective, D, and working, W, microphones sent out.
Let X be the number of defective microphone sent out. Construct the probability distribution table for X.
Find the expected value of X.
The online retailer gives gift vouchers to customers who return their defective microphones. The value of gift voucher is given by C \left( X \right) = 4 X. Calculate the expected value of the gift voucher for this batch.
In a game of two-up, a person called the “Spinner” tosses two coins.
-
If the coins land with two heads up, then the Spinner wins and the gamblers lose.
-
If the coins land with two tails up, the Spinner loses and the gamblers win.
-
If the coins land one head up and one tail up, the Spinner tosses the coins again and the gamblers break even.
Construct a tree diagram to represent all possible outcomes of tossing two coins.
If each gambler bets \$3, and can win \$3 per toss, construct a probability distribution table for the profit of the gambler for one game of two-up.
Let the set of numbers n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 be a sample space, and let L be a random variable over the sample space defined as L \left(n\right) = number of letters in the English word for the number n. So L(1) = 3 because the word "one" has three letters.
Each outcome of the sample space is equally likely to occur. Complete the probability distribution table for L:
| n | 3 | 4 | 5 | 6 |
|---|---|---|---|---|
| P(L = n) |
Find the average number of letters in the English words for numbers 1 to 12.
Consider the table:
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| P(X = x) | 0.1 | 0.45 | 0.15 | 0.05 | 0.25 |
Does the table represent a discrete probability distribution? Explain your answer.
State the most likely outcome for the random variable X.
Calculate E \left(X\right).
For each of the following probability distribution tables for the random variable X:
Find the most likely outcome.
Find the expected value, E \left (X \right).
| x | -6 | -4 | -2 | 0 | 2 | 4 | 6 |
|---|---|---|---|---|---|---|---|
| P(X = x) | \dfrac{2}{32} | \dfrac{4}{32} | \dfrac{6}{32} | \dfrac{8}{32} | \dfrac{6}{32} | \dfrac{4}{32} | \dfrac{2}{32} |
| x | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
|---|---|---|---|---|---|---|---|---|
| P(X = x) | 0.4 | 0.04 | 0.16 | 0.07 | 0.05 | 0.23 | 0.02 | 0.03 |
| x | 2 | 5 | 8 | 11 |
|---|---|---|---|---|
| P(X = x) | 0.05 | 0.15 | 0.2 | 0.6 |
| x | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 |
|---|---|---|---|---|---|---|---|---|
| P(X = x) | 0.04 | 0.03 | 0.07 | 0.35 | 0.11 | 0.08 | 0.19 | 0.13 |
Consider the following of each statements:
Write the notation for the sample mean.
Write the notation for the population mean.
Describe the relationship between a population and a sample.
Over a 12 game season Jack scored the following points: 12, 12, 18, 9, 12, 9, 11, 15, 15, 19, 15, 18
Find the population mean.
In three games against his arch rivals he averaged 10. Is \mu bigger or smaller than \overline{x}?