Liam is playing two games. He is equally likely to win each game. The probability that Liam will win at least one of the games is 80%.
Which of the following is closest to the probability that Liam will win both games?
- 31%
- 40%
- 55%
- 64%
Aussie Maths & Science Teachers: Save your time with SmarterEd
Liam is playing two games. He is equally likely to win each game. The probability that Liam will win at least one of the games is 80%.
Which of the following is closest to the probability that Liam will win both games?
`A`
`Ptext{(at least 1 W)}\ = 1-Ptext{(LL)}\ =0.8`
`Ptext{(LL)}` | `=0.2` | |
`Ptext{(L)}` | `=sqrt0.2` | |
`=0.447` |
`Ptext{(W)}` | `=1-0.447=0.553` | |
`Ptext{(WW)}` | `=(0.553)^2` | |
`=0.31` |
`=>A`
The probability that a person chosen at random has red hair is 0.02
What is the probability that at least ONE has red hair? (2 marks)
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a. | `P(R)` | `= 0.02` |
`P(barR)` | `= 0.98` |
`P\ text{(At least 1 has red hair)}`
`= 1 – P(barR, barR)`
`= 1 – 0.98 xx 0.98`
`= 0.0396`
b. `text(Find)\ \ n\ \ text(such that)`
`1 – 0.98^n` | `> 0.4` |
`0.98^n` | `< 0.6` |
`ln 0.98^n` | `< ln 0.6` |
`n ln 0.98` | `< ln 0.6` |
`n` | `> (ln 0.6)/(ln 0.98),\ \ \ (ln 0.98 <0)` |
`> 25.28…` |
`:. 26\ text(people must be chosen.)`
A game is played by tossing an ordinary 6-sided die and an ordinary coin at the same time. The game is won if the uppermost face of the die shows an even number or the uppermost face of the coin shows a tail (or both).
What is the probability of winning this game?
`C`
`text(Game lost only if an odd and a head show.)`
`:. P(W)` | `= 1 – P text{(odd)} ⋅ P text{(head)}` |
`= 1 – 3/6 ⋅ 1/2` | |
`= 3/4` |
`=> C`
Two machines, `A` and `B`, produce pens. It is known that 10% of the pens produced by machine `A` are faulty and that 5% of the pens produced by machine `B` are faulty.
What is the probability that at least one of the pens is faulty? (1 mark)
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What is the probability that neither pen is faulty? (2 marks)
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i. | `text{P(at least 1 faulty)}` | `= 1 – text{P(both faulty)}` |
`= 1 – 0.9 xx 0.95` | ||
`= 1 – 0.855` | ||
`= 0.145` |
ii. `text{P(2 non-faulty pens})`
`= text{(choose A, NF, NF)} + P text{(choose B, NF, NF)}`
`= 1/2 xx 0.9 xx 0.9 + 1/2 xx 0.95 xx 0.95`
`= 0.405 + 0.45125`
`=0.85625`
In a game, a turn involves rolling two dice, each with faces marked 0, 1, 2, 3, 4 and 5. The score for each turn is calculated by multiplying the two numbers uppermost on the dice.
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i. |
`:. P(0) = (11)/(36)`
ii. `P(≥ 16)= 4/36=1/9`
iii. `Ptext{(Sum} < 45) = 1 − Ptext{(Sum} ≥ 45)`
`Ptext{(Sum} ≥ 45)` | `=P(20,25)+P(25,20)+P(25,25)` |
`=(2/36 xx 1/36) + (2/36 xx 1/36)+(1/36 xx 1/36)` | |
`=2/1296 + 2/1296+ 1/1296` | |
`=5/1296` |
`:.Ptext{(Sum} < 45)` | `= 1 − 5/1296` |
`= 1291/1296` |
Two ordinary dice are rolled. The score is the sum of the numbers on the top faces.
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Weather records for a town suggest that:
In a specific week Thursday is dry. The tree diagram shows the possible outcomes for the next three days: Friday, Saturday and Sunday.
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i. `text{Show}\ \ P text{(Sat dry)} = 2/3`
`P text{(Sat dry)}`
`= P (W,D) + P (D, D)`
`=(1/2 xx 5/6) + (1/2 xx 1/2)`
`= 5/(12) + 1/4`
`= 2/3\ \ text(… as required)`
ii. `Ptext{(Sat and Sun wet)}`
`= P (WWW) + P (DWW)`
`= (1/2 xx 1/6 xx 1/6) + (1/2 xx 1/2 xx 1/6)`
`= 1/(72) + 1/(24)`
`= 1/(18)`
iii. `Ptext{(At least Sat or Sun dry)}`
`= 1 – Ptext{(Sat and Sun both wet)}`
`= 1 – 1/(18)`
`= (17)/(18)`
The probability that Mel’s soccer team wins this weekend is `5/7`.
The probability that Mel’s rugby league team wins this weekend is `2/3`.
What is the probability that neither team wins this weekend?
`A`
`Ptext{(win at soccer)} = 5/7`
`:. Ptext{(not win at soccer)} = 1 – 5/7 = 2/7`
`Ptext{(win at league)} = 2/3`
`:. Ptext{(not win at league)} = 1/3`
`:. Ptext{(not win at both)}` | `= 2/7 xx 1/3` |
`= 2/21` |
`=> A`
A chessboard has 32 black squares and 32 white squares. Tanya chooses three different squares at random.
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i. `text(P)(WWW)` | `= 32/64 xx 31/63 xx 30/62` |
`= 5/42` |
ii. `text{P(same colour)}`
`= P(WWW) + P(BBB)`
`= 5/42 + 32/64 xx 31/63 xx 30/62`
`= 5/42 + 5/42`
`= 5/21`
iii. `text{P(not all the same colour)}`
`= 1 – text{P(same colour)}`
`= 1 – 5/21`
`= 16/21`
A total of 300 tickets are sold in a raffle which has three prizes. There are 100 red, 100 green and 100 blue tickets.
At the drawing of the raffle, winning tickets are NOT replaced before the next draw.
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i. | `P(R R R)` | `= 100/300 xx 99/299 xx 98/298` |
`= 1617/(44\ 551)` |
ii. `Ptext{(at least 1 winner NOT red)}`
`= 1 − P(R R R)`
`= 1− 1617/(44\ 551)`
`= (42\ 934)/(44\ 551)`
iii. `text(# Combinations of winning tickets)`
`= 3 xx 2 xx 1`
`= 6`
`:.P text{(one winner from each colour)}`
`= 6 xx 100/300 xx 100/299 xx 100/298`
`= 0.22446…`
`= 0.224\ \ text{(to 3 d.p.)}`
It is estimated that 85% of students in Australia own a mobile phone.
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i. `P(M) = 0.85`
`P(M^c) = 1-0.85 = 0.15`
`:.\ P(M^c, M^c)` | `= 15/100 * 15/100` |
`= 225/(10\ 000)` | |
`= 9/400` |
ii. `text{P(owns mobile and used it)}`
`= P(M) xx P\text{(used it)}`
`= 17/20 xx 20/100`
`= 17/100`
Xena and Gabrielle compete in a series of games. The series finishes when one player has won two games. In any game, the probability that Xena wins is `2/3` and the probability that Gabrielle wins is `1/3`.
Part of the tree diagram for this series of games is shown.
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i. |
ii. `P text{(} G\ text(wins) text{)}`
`= P(XGG) + P (GXG) + P (GG)`
`= 2/3 * 1/3 * 1/3 + 1/3 * 2/3 * 1/3 + 1/3 * 1/3`
`= 2/27 + 2/27 + 1/9`
`= 7/27`
iii. `text(Method 1:)`
`P text{(3 games played)}`
`= P (XG) + P(GX)`
`= 2/3 * 1/3 + 1/3 * 2/3`
`= 4/9`
`text(Method 2:)`
`P text{(3 games)}`
`= 1 – [P(XX) + P(GG)]`
`= 1 – [2/3 * 2/3 + 1/3 * 1/3]`
`= 1 – 5/9`
`= 4/9`
Three runners compete in a race. The probabilities that the three runners finish the race in under 10 seconds are `1/4`, `1/6` and `2/5` respectively.
What is the probability that at least one of the three runners will finish the race in under 10 seconds?
`D`
`text{P} (R_1 < 10\ text(secs) ) = 1/4\ \ =>text{P} (bar R_1) = 3/4`
`text{P} (R_2 < 10\ text(secs) ) = 1/6\ \ =>text{P} (bar R_2) = 5/6`
`text{P} (R_3 < 10\ text(secs) ) = 2/5\ \ =>text{P} (bar R_3) = 3/5`
`:.\text{P} ( text(at least)\ 1 < 10\ text(secs) )`
`= 1\ -text{P} ( text(all) >= 10\ text(secs) )`
`= 1\ – 3/4 xx 5/6 xx 3/5`
`= 1\ – 45/120`
`= 5/8`
`=> D`
Each week Van and Marie take part in a raffle at their respective workplaces.
The probability that Van wins a prize in his raffle is `1/9`. The probability that Marie wins a prize in her raffle is `1/16`.
What is the probability that, during the next three weeks, at least one of them wins a prize? (2 marks)
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`91/216`
`P text{(Van loses)}` | `= 1 – 1/9 = 8/9` |
`P text{(Marie loses)}` | `= 1 – 1/16 = 15/16` |
`P text{(both lose)}` | `= 8/9 xx 15/16 = 5/6` |
`text{P(At least 1 wins)}`
`= 1\ – P text{(both lose for 3 weeks)}`
`= 1\ – (5/6)(5/6)(5/6)`
`= 1\ – 125/216`
`= 91/216`
On each working day James parks his car in a parking station which has three levels. He parks his car on a randomly chosen level. He always forgets where he has parked, so when he leaves work he chooses a level at random and searches for his car. If his car is not on that level, he chooses a different level and continues in this way until he finds his car.
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i. `P text{(1st chosen)} = 1/3`
ii. `P text{(search 3 levels)}`
`= P text{(not 1st)} xx P text{(not 2nd)}`
`= 2/3 xx 1/2`
`= 1/3`
iii. `P text{(not 1st for 5 days)}`
`= 2/3 xx 2/3 xx 2/3 xx 2/3 xx 2/3`
`= 32/243`
Two identical biased coins are tossed together, and the outcome is recorded.
After a large number of trials it is observed that the probability that both coins land showing heads is 0.36.
What is the probability that both coins land showing tails? (2 marks)
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`0.16`
`P(H_1 H_2)` | `=P(H_1) xx P(H_2)` |
`=0.36` |
`text(S)text(ince coins are identical:)`
`P(H)` | `= sqrt 0.36` |
`= 0.6` |
`P(T)` | `= 1 – P(H)` |
`=0.4` |
`:.\ P(T_1 T_2)` | `= 0.4 xx 0.4` |
`=0.16` |
There are twelve chocolates in a box. Four of the chocolates have mint centres, four have caramel centres and four have strawberry centres. Ali randomly selects two chocolates and eats them.
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i. | `P text{(2 mint)}` | `=P(M_1) xx P(M_2)` |
`=4/12 xx 3/11` | ||
`=1/11` |
ii. | `P text{(2 same)}` | `=P(M_1 M_2) + P(C_1 C_2) + P(S_1 S_2)` |
`=1/11 + (4/12 xx 3/11) + (4/12 xx 3/11)` | ||
`=3/11` |
iii. `text(Solution 1)`
`P text{(2 diff)}` | `=1\ – P text{(2 same)}` |
`=1\ – 3/11` | |
`=8/11` |
`text(Solution 2)`
`P text{(2 diff)}` | `=P(M_1,text(not)\ M_2 text{)} + P(C_1,text(not)\ C_2 text{)} + P(S_1,text(not)\ S_2 text{)}` |
`=(4/12 xx 8/11) + (4/12 xx 8/11) + (4/12 xx 8/11)` | |
`=32/121 + 32/121 + 32/121` | |
`=8/11` |
Two buckets each contain red marbles and white marbles. Bucket `A` contains 3 red and 2 white marbles. Bucket `B` contains 3 red and 4 white marbles.
Chris randomly chooses one marble from each bucket.
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i. `P{(both red)}` | `= P(R_1) xx P(R_2)` |
`= 3/5 xx 3/7` | |
`= 9/35` |
ii. `Ptext{(at least one white)}` | `= 1 – Ptext{(none white)}` |
`= 1 – P(R_1) xx P(R_2)` | |
`= 1 – 9/35` | |
`= 26/35` |
iii. `Ptext{(same colour)}` | `= P(R_1 R_2) + P(W_1 W_2)` |
`= 9/35 + (2/5 xx 4/7)` | |
`= 9/35 + 8/35` | |
`= 17/35` |
A bag contains 4 red marbles and 6 blue marbles. Three marbles are selected at random without replacement.
What is the probability that at least one of the marbles selected is red?
`C`
`text{P(at least 1 red)}` | `= 1 – Ptext{(none red)}` |
`= 1 – P(B_1) xx P(B_2) xx P(B_3)` | |
`= 1 – 6/10 xx 5/9 xx 4/8` | |
`= 1 – 120/720` | |
`= 1 – 1/6` | |
`= 5/6` |
`=> C`