10.08 With or without replacement
A pile of playing cards has 4 diamonds and 3 hearts. A second pile has 2 diamonds and 5 hearts. One card is selected at random from the first pile, then the second.
Construct a probability tree of this situation with the correct probability on each branch.
Find the probability of selecting two hearts.
In tennis if the first serve is a fault (out or in the net), the player takes a second serve. A player serves with the following probabilities:
First serve in: 0.55
- Second serve in: 0.81
Construct a probability tree showing the probability of the first two serves either being in or a fault.
Find the probability that the player needs to make a second serve.
Find the probability that the player makes a double fault (both serves are a fault).
A container holds four counters coloured red, blue, green and yellow. Draw a tree diagram representing all possible outcomes when two draws are done, and the first counter is:
Replaced before the second draw.
Not replaced before the next draw.
A container holds three cards coloured red, blue and green.
Draw a tree diagram representing all possible outcomes when two draws occur, and the card is not replaced before the next draw.
Find the probability of drawing the blue card first.
Find the probability of drawing a blue card in either the first or second draw.
Find the probability of drawing at least one blue card if the cards are replaced before the next draw.
Consider the word WOLLONGONG. If three letters are randomly selected from it without replacement, find the probability that:
The letters are W, O, L, in that order.
The letters are O, N, G, in that order.
All three letters are O.
None of the three letters is an O.
Two marbles are randomly drawn without replacement from a bag containing 1 blue, 2 red and 3 yellow marbles.
Construct a tree diagram to show the sample space.
Find the probability of drawing the following:
A blue marble and a yellow marble, in that order.
A red marble and a blue marble, in that order.
2 red marbles.
No yellow marbles.
2 blue marbles.
A yellow marble and a red marble, in that order.
A yellow and a red marble, in any order.
There are 4 green counters and 8 purple counters in a bag. Find the probability of choosing a green counter, not replacing it, then choosing a purple counter.
A standard deck of cards is used and 3 cards are drawn out. Find the probability, in fraction form, of the following:
All 3 cards are diamonds if the cards are drawn with replacement.
All 3 cards are diamonds if the cards are drawn without replacement.
A number game uses a basket with 6 balls, all labelled with numbers from 1 to 6. 2 balls are drawn at random. Find the probability that the ball labelled 3 is picked once if the balls are drawn:
With replacement.
Without replacement.
A hand contains a 10, a jack, a queen, a king and an ace. Two cards are drawn from the hand at random, in succession and without replacement. Find the probability that:
The ace is drawn.
The king is not drawn.
The queen is the second card drawn.
Eileen randomly selects two cards, with replacement, from a normal deck of cards. Find the probability that:
The first card is a queen of spades and the second card is a 4 of clubs.
The first card is spades and the second card is a 4.
The first card is a Queen and the second card is black.
The first card is not a 7 and the second card is not Clubs.
Find the probability of drawing a green counter from a bag of 5 green counters and 6 black counters, replacing it and drawing another green counter.
A chess player is placed into a draw where in each match he has a 30\% chance of winning. Find the probability that:
He wins his first two matches.
He wins his first three matches.
He wins his first two matches and then loses his third match.
Beth randomly selects three cards, with replacement, from a normal deck of cards. Find the probability that:
The cards are queen of diamonds, king of spades, and king of diamonds, in that order.
The cards are all black.
The first card is a 9, the second card is a heart and the third card is red.
The cards are all hearts.
None of the cards is a 9.
From a standard pack of cards, one card is randomly drawn and then put back into the pack. A second card is then drawn. Calculate the probability that:
Neither of the cards are diamonds
At least one of the cards is a diamond
Three marbles are randomly drawn with replacement from a bag containing 6 red, 4 yellow, 3 white, 2 black and 4 green marbles. Find the probability of drawing:
Three white marbles
No green marbles
At least one red marble
At least one white marble
Amelia randomly selects two cards, with replacement, from a normal deck of cards. Find the probability that:
Both cards are red
Both cards are the same colour
Both cards are of different colours
A bag contains four marbles - red, green, blue and yellow. Beth randomly selects a marble, returns the marble to the bag and selects another marble.
Construct a tree diagram for the experiment given.
Find the probability of Beth selecting:
A blue and a yellow marble.
A blue followed by a yellow marble.
2 red marbles.
2 marbles of the same colour.
2 marbles of different colours.
James, a test cricketer, analysed his past innings and found his probabilities of scoring particular numbers of runs, as shown in the table:
| \text{Number of runs} | 0 | 1 - 20 | 21 - 49 | 50 - 99 | 100+ |
|---|---|---|---|---|---|
| \text{Probability} | \dfrac{1}{10} | \enspace \enspace \dfrac{3}{10} | \enspace \enspace \dfrac{3}{10} | \enspace \enspace \dfrac{2}{10} | \enspace \dfrac{1}{10} |
If James is selected to play in the next test match, calculate the probability that he scores:
0 runs in exactly one of the two innings.
0 runs in both innings.
At least 50 runs in at least one of the innings.
A total of 100 or more runs in each of the two innings.
More than 20 runs in only one of the two innings.
Three cards labeled 1, 2, 3 are placed face down on a table. Two of the cards are selected randomly to form a two-digit number. The possible outcomes are displayed in the following probability tree:
List the sample space of two digit numbers produced by this process.
Find the probability that 2 is a digit in the number.
Find the probability that the sum of the two selected cards is even.
Find the probability of forming a number greater than 40.
In a school, 25\% of students ride skateboards and 20\% of students have dark hair. One student is selected at random. Find the probability that the student:
Has dark hair and rides a skateboard.
Has light hair and does not ride a skateboard.
Has dark hair and does not ride a skateboard.
Has light hair and rides a skateboard.
The ratio of left-handed people to right-handed people in a country is 4:3. Two people are surveyed at random. Calculate the probability that:
Both people are left-handed.
One person is left-handed and the other is right-handed.
At least one person is right-handed.