Theoretical Probability
A fair coin is tossed 20 times. What is the theoretical probability of obtaining heads? Simplify your answer.
P(H) = 10/20 = 1/2
A die is rolled 10 times. A frequency table of the outcomes is shown below.
Outcome 1 | 2 | 3 | 4 | 5 | 6
Frequency 3 | 1 | 1 | 1 | 0 | 4
What is the relative frequency (experimental probability) of rolling a 6? Write your answer in simplest form.
P(6) = 4/10 = 2/5
Use the two-way table below to determine how many students were surveyed.
| Soccer | Tennis |
Boys | 5 | 20 | 25
Girls | 10 | 3 | 13
| 15 | 23 | 38
38 total.
A dart is thrown at a dartboard 200 times and it landed in the scoring area 163 times and outside the scoring area for the remiander.
Estimate the probability that when two darts are thrown at the board, both are scoring shots.
P(hitting the scoring area) = 163/200
P(two darts hit)
= (163/200) x (163/200)
= 0.66 or 26569/40000
The sum of all probabilities (theoretical or experimental) should equal ___?
1
If event (A) is your test falling on a Friday, list the complement of that event.
A' = Monday, Tuesday, Wednesday, Thursday
(assuming no tests on Saturday or Sunday)
Using the table, what is the probability that a randomly selected student is a boy who likes tennis?
| Soccer | Tennis |
Boys | 5 | 20 | 25
Girls | 10 | 3 | 13
| 15 | 23 | 38
20/38
= 10/19
A deck of 52 playing cards is shuffled. Two cards are drawn, one after the other, and the colour each card is noted. Draw a tree diagram with probabilities clearly marked.
Teacher to draw correct answer on the board.
Options for colours are Black (B) and Red (R)
School X's debate team consists of Adam, Bob and Claire. School Y's team consists of David, Emma and Felicity. One member of each team is randomly chosen to be the first speaker.
Use a grid to illustrate the sample space.
Adam AD AE AF
Bob BD BE BF
Claire CD CE CF
David Emma Felicity
A spinner numbered 1-4 was spun a certain number of times and the following results were achieved.
Outcome 1 | 2 | 3 | 4
Frequency 7 | 26 | 9 | 58
How many trials were conducted?
100 trials
(add all the frequencies 7+26+9+58=100)
What is the probability that a randomly chosen student is right-handed, given that the student is male?
| Right Handed | Left Handed |
Female | 58 | 13 | 71
Male | 47 | 12 | 59
| 105 | 25 | 130
47/59
A bag containing mixed nuts is shared among friends. There are 10 macadamias, 5 cashews, and 15 peanuts. What is the probability of the first and second friend both choosing and eating a peanut?
This is without replacement.
P(Peanut and Peanut)
= (15/30) x (14/29)
= 7/29
= 0.241
A fair die is rolled. What is the probability of obtaining a number less than 5? Simplify your answer.
P(<5)=P(1, 2, 3 or 4) = 4/6 = 2/3
At a birthday party, some cans of soft drink were put in a container of ice. There were 16 cans of Coke, 20 cans of Sprite, 13 cans of Fanta, 8 cans of Sunkist and 15 cans of Pepsi.
If a can was picked at random, what is the probability that it was not a can of Fanta?
59/72
Total number of cans = 16+20+13+8+15=72
Cans that are not fanta = 16+20+8+15 = 59
Note:
P(not fanta) = 1 - P(fanta) = 1-(13/72) = 59/72
Use the table below to determine the probability that a randomly selected student is a girl given they are a right handed.
| Left Handed | Right Handed |
Boys | 17 | | 35
Girls | | |
| 29 | | 70
23/41
| Left Handed | Right Handed |
Boys | 17 | 18 | 35
Girls | 12 | 23 | 35
| 29 | 41 | 70
A bag contains 4 red and 6 yellow balls. If the first ball drawn is yellow, find the probability of drawing a second yellow ball given that the first ball was not replaced.
5/9
Note that the first ball has already been drawn. There are only 9 balls left to choose from overall (originally there were 10). Since the first ball was yellow, there are only 5 yellow balls left in the bag.
A coin is biased so that the chance of it falling as a Head when flipped is 0.75. What is the probability of getting 1 heads and 2 tails if it is tossed 3 times? (in any order)
P(1H and 2T)
= P(HTT) + P(TTH) + P(THT)
= P(0.75x0.25x0.25) + P(0.25x0.25x0.75) + (0.25x0.75x0.25)
= 0.047+0.047+0.047
= 3 x 0.047
= 0.141
A coin is tossed multiple times. A head is recorded 7 times and a tail 3 times.
Find the difference between the experimental and theoretical probability of tossing a head.
Experimental probability = 7/10
Theoretical probability = 5/10
Difference = 7/10 - 5/10 = 2/10 = 1/5
There are 120 students in Year 9. 55% of them are boys. One-third of the girls don't play footy. 30 students in the entire year level don't play footy. Represent this information in a two-way table.
| Girls | Boys |
Footy | 36 | 54 | 90
No Footy | 18 | 12 | 30
| 54 | 66 | 120
55% of 120 = 0.55 x 120 = 66 boys
1/3 of girls = 54/3 = 18
A raffle has tickets numbered 1 to 100. There are three winnning tickets which are drawn at random, without replacement. Find the probability that all the numbers drawn are even.
This is without replacement.
P(even, even and even)
=(50/100) x (49/99) x (48/98)
= 4/33
There are 50 even numbers from 1 to 100. So the probability of choosing an even number on the first pick is 50/100.
The probability of choosing an even number on the second pick is 49/99 (since there are 49 even numbers left and 99 numbers left overall).
The probability of choosing an even number on the third pick is 48/98 (since there are 48 even numbers lefts and 98 numbers left overall).