Probability and Statistics: practice solutions, Grade 11 – download the PDF

1 Lunch combinations ★★★

Three stages, so we multiply: \(5\times4\times3=60\).

Answer: 60 different lunches.

2 Factorial arithmetic ★★★

\(6!=720\).

\(\dfrac{8!}{6!}=8\times7=56\).

\(P(7,2)=7\times6=42\).

\(C(7,2)=\dfrac{42}{2!}=21\).

3 Bike lock ★★★

(a) \(10\times10\times10\times10=10{,}000\).

(b) \(10\times9\times8\times7=5{,}040\).

4 Pizza toppings ★★★

Order does not matter (the same toppings give the same pizza), so we use a combination: \(C(9,3)=\dfrac{9\times8\times7}{3\times2\times1}=84\).

Answer: 84 pizzas.

5 Coin and die ★★★

The coin and the die do not influence each other, so the events are independent: \(\dfrac12\times\dfrac36=\dfrac14\).

Answer: \(\dfrac14=0.25\).

6 Tea or coffee ★★★

(a) There are \(9+21=30\) people under 30: \(P=\dfrac{21}{30}=0.7\).

(b) There are \(11+19=30\) people 30 or older: \(P=\dfrac{11}{30}\approx0.367\).

7 Heights and the empirical rule ★★★

(a) \(66\pm3\): between 63 and 69 inches.

(b) \(66\pm6\): between 60 and 72 inches.

(c) 72 inches is 2 standard deviations above the mean. About 95% lie within, so the two tails share 5%, and the upper tail is about \(2.5\%\).

8 Books on a shelf ★★★

Glue the two books into one block. Then we arrange 4 objects: \(4!=24\) ways. Inside the block the two books can be swapped: \(2!=2\) ways.

Total: \(24\times2=48\).

9 Choosing a committee ★★★

Choose the girls: \(C(4,2)=6\). Choose the boys: \(C(5,2)=10\). Multiply the two stages: \(6\times10=60\).

10 Two aces ★★★

The draws are dependent: \(\dfrac4{52}\times\dfrac3{51}=\dfrac{12}{2652}=\dfrac1{221}\).

Answer: \(\dfrac{1}{221}\approx0.0045\).

11 Exactly one red ★★★

Two ways: red then green, or green then red.

\(\dfrac6{10}\times\dfrac49+\dfrac4{10}\times\dfrac69=\dfrac{24}{90}+\dfrac{24}{90}=\dfrac{48}{90}=\dfrac8{15}\).

12 Guessing on a quiz ★★★

Binomial with \(n=8\), \(p=0.5\): \(P(X=6)=C(8,6)(0.5)^6(0.5)^2=\dfrac{28}{256}=\dfrac7{64}\approx0.109\).

13 Expanding a binomial ★★★

Row 4 of Pascal’s triangle: \(1,4,6,4,1\). Powers of \(-3\): \(1,-3,9,-27,81\).

\((x-3)^4=x^4-12x^3+54x^2-108x+81\).

14 Standard deviation by hand ★★★

Mean: \(\dfrac{32}{8}=4\).

Squared deviations: \(9,1,1,1,0,0,4,16\), sum \(32\).

\(\sigma=\sqrt{32/8}=\sqrt4=2\).

15 Which test was better? ★★★

Maya: \(z=\dfrac{82-70}{8}=1.5\).

Brother: \(z=\dfrac{31-25}{3}=2\).

His score is 2 standard deviations above the mean, Maya’s is 1.5. Relative to his class, the brother did better.

16 Stratified sample ★★★

Total: \(240+220+200+140=800\), so the sampling fraction is \(\dfrac{80}{800}=10\%\).

9th: 24; 10th: 22; 11th: 20; 12th: 14. Check: \(24+22+20+14=80\).

17 Letters of STATISTICS ★★★

There are 10 letters: S three times, T three times, I twice, A once, C once.

\(\dfrac{10!}{3!\,3!\,2!}=\dfrac{3{,}628{,}800}{72}=50{,}400\).

18 A screening test ★★★

Sick: \(0.02\times1000=20\) people, of whom \(0.9\times20=18\) test positive.

Healthy: 980 people, of whom \(0.05\times980=49\) test positive.

\(P(\text{sick}\mid\text{positive})=\dfrac{18}{18+49}=\dfrac{18}{67}\approx0.269\).

Only about 27%: because the condition is rare, most positives are false alarms.

19 Defective parts ★★★

(a) Complement: \(1-0.9^{10}\approx1-0.3487=0.6513\).

(b) \(C(10,2)(0.1)^2(0.9)^8=45\times0.01\times0.4305\approx0.1937\).

(c) Mean \(=10\times0.1=1\). Standard deviation \(=\sqrt{10\times0.1\times0.9}=\sqrt{0.9}\approx0.949\).

20 Coefficients ★★★

(a) The term with \(x^3\) is \(C(6,3)(2x)^3(1)^3=20\times8x^3=160x^3\). The coefficient is 160.

(b) Replace \(x\) by 1: \((2+1)^6=3^6=729\).

21 Polling margin of error ★★★

(a) \(\sqrt{\dfrac{0.64\times0.36}{625}}=\dfrac{0.48}{25}=0.0192\), so \(ME=1.96\times0.0192\approx0.0376\), about 3.8%.

(b) \(0.64\pm0.0376\): from about 60.2% to 67.8%.

(c) \(1.96\times\dfrac{0.5}{\sqrt n}\le0.03\) gives \(\sqrt n\ge32.67\), so \(n\ge1067.1\). A sample of 1,068 residents is needed.

22 Battery life ★★★

(a) 30 and 50 are \(\mu\pm2\sigma\): about 95%.

(b) 45 is \(\mu+\sigma\). Outside \(\pm\sigma\) lies 32%, half of it above: about 16%.

(c) From \(-1\sigma\) to \(+2\sigma\): \(34\%+47.5\%=81.5\%\). Then \(0.815\times2500=2037.5\), about 2,038 batteries.