Aussie Maths & Science Teachers: Save your time with SmarterEd
In a game, the probability that a particular player scores a goal at each attempt is 0.15.
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a. \(0.7225\)
b. \(n=10\)
a. \(P(G)=0.15, \ \ P(\bar{G})=0.85\)
\(P(\bar{G}\bar{G}) = 0.85^2=0.7225\)
b. \(\text{2 attempts:}\ P(\text{at least 1 goal})=1-P(\bar{G}\bar{G})=1-0.85^{2}\)
\(\text{3 attempts:}\ P(\text{at least 1 goal})=1-0.85^{3}\)
\(\text{n attempts:}\ P(\text{at least 1 goal})=1-0.85^{n}\)
\(\text{Find}\ n\ \text{such that:}\)
| \(1-0.85^{n}\) | \(\gt 0.8\) | |
| \(0.85^{n}\) | \(\lt 0.2\) | |
| \(n \times\ln(0.85)\) | \(\lt \ln(0.2)\) | |
| \(n\) | \(\gt \dfrac{\ln(0.2)}{\ln(0.85)}\) | |
| \(\gt 9.9…\) |
\(\therefore\ \text{Least}\ n=10\)
A box contains \(n\) green balls and \(m\) red balls. A ball is selected at random, and its colour is noted. The ball is then replaced in the box.
In 8 such selections, where \(n\neq m\), what is the probability that a green ball is selected at least once?
\(C\)
\(\text{In any single selection:}\)
\(P(\text{green})\ =\Bigg(\dfrac{n}{n+m}\Bigg), \ \ P(\text{not green})\ =\Bigg(\dfrac{m}{n+m}\Bigg) \)
\(\text{Let}\ \ X=\ \text{choosing a green ball}\)
| \(P(X\geq 1)\) | \(=1-\text{Pr}(X=0)\) |
| \(=1-\Bigg(\dfrac{m}{n+m}\Bigg)^8\) |
\(\Rightarrow C\)
In a bag there are 3 six-sided dice. Two of the dice have faces marked 1, 2, 3, 4, 5, 6. The other is a special die with faces marked 1, 2, 3, 5, 5, 5.
One die is randomly selected and tossed.
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| a. | `P(5)` | `=1/3 xx 1/6 + 1/3 xx 1/6 + 1/3 xx 1/2` |
| `=1/18+1/18+1/6` | ||
| `=5/18` |
b. `Ptext{(special die given a 5 is rolled)}`
`=(Ptext{(special die)} ∩ P(5))/(P(5))`
`=(1/3 xx 1/2)/(5/18)`
`=1/6 xx 18/5`
`=3/5`
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`
At the start of a particular week, Kim has three red apples and two green apples. She eats one apple everyday. On Monday, Tuesday and Wednesday of that week, she randomly selects an apple to eat. In this three-day period, the probability that Kim does not eat an apple of the same colour on any two consecutive days is
`B`
`P text{(alternate colours)}`
`= P(RGR) + P(GRG)`
`= (3)/(5) ·(2)/(4) ·(2)/(3) + (2)/(5) ·(3)/(4) ·(1)/(3)`
`= (12)/(60) + (6)/(60)`
`= (3)/(10)`
`=> \ B`
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})`♦ Mean mark 48%.
`= 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`
A runner has four different pairs of shoes.
If two shoes are selected at random, what is the probability that they will be a matching pair?
`C`
`text(Strategy One:)`♦♦ Mean mark 31%.
`text(Choose 1 shoe then find the probability)`
`text(the next choice is matching).`
| `P` | `= 1 xx 1/7` |
| `= 1/7` |
| `P` | `= text(Number of desired outcomes)/text(Number of possibilities)` |
| `= 4/(\ ^8C_2)` | |
| `= 4/28` | |
| `= 1/7` |
`=> C`
An eight- sided die is marked with numbers 1, 2, … , 8. A game is played by rolling the die until an 8 appears on the uppermost face. At this point the game ends.
`qquad qquad 1/8 + 7/8 xx 1/8 + (7/8)^2 xx 1/8`. (2 marks)
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i. `P text{(game ends before 4th roll)}`
`= P (8) + P (text{not}\ 8, 8) + P (text{not}\ 8, text{not}\ 8, 8)`
`= 1/8 + 7/8 · 1/8 + 7/8 · 7/8 · 1/8`
`= 1/8 + 7/8 · 1/8 + (7/8)^2 · 1/8\ \ text(… as required)`
ii. `1/8 + 7/8 · 1/8 + (7/8)^2 · 1/8 + …`
`=> text(GP where)\ \ a = 1/8,\ \ r = 7/8`
`text(Find)\ \ n\ \ text(such that)\ \ S_(n – 1) > 3/4,`
| `S_(n-1)` | `= (a (1 – r^(n – 1)))/(1 – r)` |
| `3/4` | `< 1/8 xx {(1 – (7/8)^(n – 1))}/(1 – 7/8)` |
| `3/4` | `< 1 – (7/8)^(n – 1)` |
| `(7/8)^(n – 1)` | `< 1/4` |
| `(n-1)* ln\ 7/8` | `< ln\ 1/4` |
| `n – 1` | `> (ln\ 1/4)/(ln\ 7/8)` |
| `> 11.38…` |
`:. n = 12`
A pack of 52 cards consists of four suits with 13 cards in each suit.
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i. `text(After 1st card is drawn)`
`text(# Cards left from another suit) = 39`
`text(# Cards left in pack) = 51`
`:. P\ text{(2nd card from a different suit)}`
`= 39/51`
`= 13/17`
ii. `P\ text{(all 4 cards from different suits)}`
`= 52/52 xx 39/51 xx 26/50 xx 13/49`
`= 2197/(20\ 825)`
`= 0.1054…`
`= 0.105\ \ \ text{(to 3 d.p.)}`
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.)}`
A packet contains 12 red, 8 green, 7 yellow and 3 black jellybeans.
One jellybean is selected from the packet at random.
What is the probability that the selected jellybean is red or yellow? (2 marks)
`19/30`
`text(12 R, 8 G, 7 Y, 3 B)`
`text(Total jellybeans) = 30`
| `P text{(R or Y)}` | `=\ text{(# Red + Yellow)}/text(Total jellybeans)` |
| `= (12 + 7)/30` | |
| `= 19/30` |
It is estimated that 85% of students in Australia own a mobile phone.
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i. `P(M) = 0.85`COMMENT: `M^c` is syllabus notation for the complement of event `M`.
`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`
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`
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` |
Pat and Chandra are playing a game. They take turns throwing two dice. The game is won by the first player to throw a double six. Pat starts the game.
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i. `P\ text{(Pat wins on 1st throw)}=P(W)`
| `P(W)` | `=P\ text{(Pat throws 2 sixes)}` |
| `=1/6 xx 1/6` | |
| `=1/36` |
ii. `text(Let)\ P(L)=P text{(loss for either player on a throw)}=35/36`
`P text{(Pat wins on 1st or 2nd throw)}`
`=P(W) + P(LL W)`
`=1/36\ + \ (35/36)xx(35/36)xx(1/36)`
`=(2521)/(46\ 656)`
`=0.054\ \ \ text{(to 3 d.p.)}`
iii. `P\ text{(Pat wins eventually)}`
`=P(W) + P(LL\ W)+P(LL\ LL\ W)+ … `
`=1/36\ +\ (35/36)^2 (1/36)\ +\ (35/36)^2 (35/36)^2 (1/36)\ +…`
`=>\ text(GP where)\ \ a=1/36,\ \ r=(35/36)^2=(1225)/(1296)`
`text(S)text(ince)\ |\ r\ |<\ 1:`
| `S_oo` | `=a/(1-r)` |
| `=(1/36)/(1-(1225/1296))` | |
| `=1/36 xx 1296/71` | |
| `=36/71` |
`:.\ text(Pat’s chances to win eventually are)\ 36/71`.