State whether X is discrete in the following:
X is a random variable for the length of agoanna.
X is a random variable for the number of games won by Manchester United in a season.
X is a random variable for the number of television a person owns.
X is a random variable for the weight of a hippopotamus to the nearest kilogram.
X is a random variable for the length of a queue at a supermarket checkout.
X is a random variable for the number of pears on each pear tree in an orchard.
A coin is tossed three times and the total number of tails is recorded.
Can this experiment be represented by a discrete random variable?
If X represents the number of possible tails in the three coin tosses, list all the possible outcomes of the experiment.
The weights of babies born in a local hospital in the last month have been recorded. One midwife is interested in the probability that of the next 5 babies born, what number of babies would weigh more than 2.4\text{ kg}.
Can this situation be modelled by a discrete random variable?
If Y represents the number of babies in the next 5 babies born that weigh more than 2.4 kg, list all the possible outcomes.
Explain why the following situations cannot be represented by a discrete random variable:
On a popular TV cooking show, contestants are to randomly choose a blue, red or yellow serviette from a box to split themselves into random cooking teams.
The weights of babies born in a local hospital in the last month have been recorded. One midwife is interested in the probability that the next baby born would weigh more than 2.8\text{ kg}.
A random number generator generates a real number between 1 and 7 inclusive.
On average, a fax machine in a busy school breaks down twice per fortnight.
Can this data be represented by a discrete random variable?
On average, the time between breakdowns of the fax machine is 72 hours. Can this data be represented by a discrete random variable? Explain your answer.
A manager randomly selects three of his staff to attend a leadership conference. He randomly selects people from his Sales team and his Development team.
Can the number of Sales people chosen be modelled by a discrete probability distribution?
Given that there are four people on the Sales team, list all of the possible outcomes of this distribution.
A teacher randomly selects 3 students from her class of 28 students to prepare a presentation for assembly. Can the number of students chosen to prepare a presentation be modelled by a probability distribution? Explain your answer.
State whether the following are examples of probability distributions:
| x | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| p (x) | 0.2 | 0.4 | 0.6 | 0.8 |
| x | 2 | 4 | 6 | 8 | 10 |
|---|---|---|---|---|---|
| p(x) | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 |
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| p(x) | -0.4 | -0 .3 | 0.8 | 0.9 |
| x | 10 | 20 | 30 | 40 |
|---|---|---|---|---|
| p(x) | 10\% | 20\% | 25\% | 45\% |
A coin is weighted such that the probability of a tail appearing uppermost is 70\%. Let X represent the number of tails appearing uppermost in two tosses of the coin.
Construct a tree diagram for this situation.
Hence, complete the table:
State the three conditions that must be true for this to be a discrete probability distribution.
| x | 0 | 1 | 2 |
|---|---|---|---|
| P(X = x) |
Hence, does this represent a discrete probability distribution?
The probability that a particular biased coin lands on tails is 0.7. Let X be the number of tails when the coin is tossed twice. Complete the given probability distribution table for X.
| x | 0 | 1 | 2 |
|---|---|---|---|
| P(X = x) |
Three marbles are randomly drawn from a bag containing seven black and three green marbles. Let X be the number of black marbles drawn.
Complete the probability distribution table for X if the marbles are drawn with replacement?
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P(X = x) |
Complete the probability distribution table for X if the marbles are drawn without replacement?
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| P(X = x) |
Two fair spinners, A and B, are spun. The number from each spinner is noted and the total score is defined below:
If X represents a discrete random variable, list all possible outcomes, x.
Hence construct the probability distribution for X.
A regular six-sided dice has a side length of 8\text{ cm}. The dice is rolled on the ground and the height above ground of the dot on the face with only a single dot is noted. Let H be the number of centimetres this single dot is above the ground.
List the possible outcomes for H.
Hence construct the probability distribution table for X.
Two regular fair dice are rolled. Let Y be the sum of the dots on the uppermost faces. Construct the probability distribution table for Y.
Does the following table represent a discrete probability distribution? Explain your answer.
| x | 2 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|
| P (X = x) | 0.1 | 0.25 | 0.3 | 0.15 | 0.2 |
A random variable X can take any of the values 0, 1, 2 or 3. The following facts about the distribution of X are known:
Use these facts to construct a probability distribution table for X.
- P \left( X = 0 \right) = \dfrac{1}{2} P \left( X = 1 \right)
- P \left( X=1 \right) = 4 P \left( X=2 \right)
- P \left( X=2 \right) = \dfrac{1}{5} P \left( X=3 \right)
The mass of each egg in a 12-pack carton of Extra Large Eggs ranges in weight from 66\text{ g} to 72\text{ g}.
Can the mass of the eggs in the carton be modelled by a continuous probability distribution?
Draw a graph that best models the shape of this continuous probability distribution.
Consider the following graph:
Identify which conditions for a discrete probability distribution are evident in the graph.
Therefore, does this graph represent a discrete probability distribution?
Fatia records her team's winning or losing margins over 10 games of the hockey season, with winning margins recorded as positive values and losing margins as negative values. The margins were recorded below:
| X | -1 | 4 | 2 | 3 | 1 | 2 | 4 | -1 | 2 | 1 |
|---|
Construct a discrete probability distribution table for X.
Four cards are randomly selected without replacement from a standard pack of 52 cards.
Can the number of clubs selected be modelled by a discrete probability distribution?
Is this distribution uniform or non-uniform?
A fair standard die is thrown. Let X be the number of dots on the uppermost face.
Construct the probability distribution table for X.
Is the discrete probability distribution uniform or non-uniform?
A dog has three puppies. Let F represent the number of female puppies in this litter.
Represent all possible outcomes of male and female puppies born using a tree diagram.
Construct the probability distribution table for F.
Is the discrete probability distribution uniform or non-uniform?
A fair standard die is rolled and the number of dots on the visible faces (the faces which are not on the ground) is noted. Let W be the number of dots that can be seen on the visible faces.
Construct the probability distribution table for W.
Is the discrete probability distribution uniform or non-uniform?
Several boxes of fruit delivered to an office building were sampled. It was found that the number of fruit in each box wasn't always the same. The frequency of each observation is given in the table:
Let X be the number of fruit in a box. Complete a probability distribution table for X using the observations.
What is the probability that the next box delivered contains at least 40 fruit?
| Number of fruit | Frequency |
|---|---|
| 38 | 5 |
| 39 | 7 |
| 40 | 24 |
| 41 | 8 |
| 42 | 26 |
What is the probability that the next box delivered contains less than 41 fruit?
Caitlin has 8 pairs of gloves in her drawer. Each pair of gloves aren't necessarily placed together. Each morning she randomly choses two gloves from the drawer in the dark to take to work. Let X be the number of left-hand gloves chosen.
Complete the probability distribution table for X.
Find the probability that the two gloves drawn form a pair.
| x | 0 | 1 | 2 |
|---|---|---|---|
| P(X = x) |
Two earrings are taken without replacement from a draw containing 3 black earrings and 5 brown earrings. Let X be the number of black earrings drawn.
Construct a probability distribution for X.
Given that at least one black earring was selected, what is the probability that two were selected?
Let X be the sum of the numbers on a pair of dice when they are rolled once.
Construct the probability distribution table for X.
Find P \left( X \leq 6 \right).
Find P\left(X \geq 4 | X \leq 6 \right).
Consider the following probability distribution table for X.
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| P(X = x) | 0.24 | a | 0.15 | b | 0.22 |
Find the value of a given that P\left(X<3\right) = 0.62.
Hence determine the value of b.
A fair die is rolled. Let X be the number it lands on.
Construct a probability distribution table for X.
Find P \left( X \geq 5 \right).
Find P \left( X \geq 5 | X \leq 5 \right).
Consider the following probability distribution table forX:
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| P(X = x) | k | k | k | k | k |
Find the value of k.
Find P \left( X > 3 \right).
Find P \left(1 \leq X < 5 \right).
Find P \left( X > 2 | X > 1 \right).
Find P \left( X \leq 4 | X \geq 2 \right).
3 marbles are randomly drawn, with replacement, from a bag containing 7 grey and 6 blue marbles. Let X be the number of grey balls drawn.
Find P\left(X = 3\right).
Find P\left(X = 2\right).
Find P\left(X \geq 2\right).
Find P\left(X = 2 | X \leq 2 \right).
The Chickens and the Doctors are playing in the grand final series of a basketball tournament in which the first team to win three games wins the series. The teams are so evenly matched that they are equally likely to win any game. Let X be the number of games that will have been played by the end of the series.
Find P \left( X = 3 \right)
Find P \left( X = 4 \right)
Find P \left( X = 5 \right)
A random variable X has the following probability distribution:
| x | 5 | 6 | 7 | 8 | 9 | 12 |
|---|---|---|---|---|---|---|
| P(X = x) | \dfrac{1}{16} | \dfrac{1}{16} | \dfrac{3}{16} | \dfrac{1}{4} | \dfrac{5}{16} | \dfrac{1}{8} |
Find P ( X = 8 )
Find P\left(X \text{ is even} \right)
Find P ( X > 8 )
Find P ( X \leq 7 )
Find P (6 < X < 8)
Find P ( 6 \leq X < 12)
Two dice are rolled and the absolute value of the differences between the numbers appearing uppermost are recorded.
Complete the sample space in the given table.
Let X be defined as the absolute value of the difference between the two dice. Construct the probability distribution table for X.
Calculate P ( X < 3 )
Calculate P ( X \leq 4 | X \geq 2)
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 0 | 3 | ||||
| 2 | 1 | |||||
| 3 | 2 | |||||
| 4 | ||||||
| 5 | 4 | 2 | ||||
| 6 |
A six-sided die with numbers from 1 to 6 is weighted such that P \left(\text{ prime number }\right) = 0.1 and P \left( 4 \right) = P\left(6\right) = 0.3. Let X represent the possible outcomes from one roll of the dice.
Construct the probability distribution table for X.
Calculate:
Consider a normal and fair nine-sided die. We are interested in how long it takes for a 5 to appear uppermost on the die for the first time.
State the probability of:
Rolling a 5 in one roll of the die.
Requiring two rolls of the die before we see a 5.
Requiring three rolls of the die before we see a 5.
Requiring 6 rolls of the die before we see a 5.
Let X be the number of rolls of the die required to see a 5 for the first time. State the conditions that are required for a discrete probability distribution for X.
Hence, does this represent a discrete probability distribution?
Consider the following probability distribution table for X:
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| P(X = x) | 0.16 | a | b | c | 0.17 |
Find the value of c given that P\left(X \geq4\right) = 0.39.
Find the value of a given that P\left(X\geq 2 | X<3\right) = 0.6.
Hence determine the value of b.