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Geometric Probability and Conditional Probability: math test solutions, Grade 10 – download the PDF

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Test solutions Grade 10 : Geometric Probability and Conditional Probability — Zyro the alien explorer of Planète Maths

Test solutions with the detailed point scale. Add up your points and spot what to review.

Suggested time: 45 minutes. Out of 20 points. Calculator allowed only when the problem says so.

1 Counting / 3 pts

(a) Order matters: \({}_9P_2=9\cdot8=72\). (1 pt)

(b) Order does not matter: \({}_9C_3=\dfrac{9\cdot8\cdot7}{6}=84\). (1 pt)

(c) In (a) the roles are different, so “Ann then Bo” and “Bo then Ann” are different outcomes; in (b) a team is the same whatever the order. (1 pt)

2 Geometric probability / 4 pts

(a) Circle: \(100\pi\). Board: \(40\cdot30=1200\). \(P=\dfrac{100\pi}{1200}=\dfrac{\pi}{12}\approx0.262\). (2 pts)

(b) \(1-\dfrac{\pi}{12}\approx0.738\). (1 pt)

(c) The cycle lasts \(40+5+35=80\) s, so \(P=\dfrac{35}{80}=\dfrac7{16}\). (1 pt)

3 Walking or riding / 4 pts

(a) Walks: 48; rides: 72; Grade 9: 60; Grade 10: 60. (1 pt)

(b) \(P(\text{walks})=\dfrac{48}{120}=\dfrac25\) (0.5 pt). \(P(\text{walks}\mid\text{Grade 10})=\dfrac{30}{60}=\dfrac12\) (1 pt).

(c) \(P(\text{walks}\cap\text{Grade 10})=\dfrac{30}{120}=0.25\), while \(\dfrac25\cdot\dfrac12=0.2\). They differ, so the events are not independent. (1.5 pts)

4 Union and complement / 3 pts

(a) \(0.55+0.4-0.25=0.7\). (1 pt)

(b) \(1-0.7=0.3\). (1 pt)

(c) \(\dfrac{0.25}{0.4}=0.625\). (1 pt)

5 Spinner and die / 3 pts

(a) \(0.3\cdot0.5=0.15\). (1 pt)

(b) \(0.3+0.5-0.15=0.65\). (1 pt)

(c) \(1-0.7^2=1-0.49=0.51\). (1 pt)

6 Draws without replacement / 3 pts

(a) \(\dfrac4{10}\cdot\dfrac39=\dfrac{12}{90}=\dfrac2{15}\). (1.5 pts)

(b) After a yellow is removed, 4 of the 9 balls are green: \(\dfrac49\). (0.5 pt)

(c) \(2\cdot\dfrac4{10}\cdot\dfrac69=\dfrac{48}{90}=\dfrac8{15}\). (1 pt)

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