Friday, January 4, 2019

Probability (Edexcel Mathematic B)

$\def\D{\displaystyle}$
1.[Edexcel 2012,winter,mathematic B, Paper 01, no 21]
 A jar contains 3 red sweets, 4 blue sweets and 7 yellow sweets. One sweet is taken, at random, from the jar and not replaced. Another sweet is then taken, at random, from the jar. A tree diagram representing these two events is shown below.

(a) Complete the tree diagram representing these two events. (2)
(b) Find the probability that both sweets are red. Give your answer as a simplified fraction.
..............................................................(2)

2[Edexcel 2012,winter, mathematic B, Paper 02, no 5]

Figure 2 shows a diagram of routes to a factory. There are four road junctions labelled $\D A, B, C$ and $\D D$ and four towns labelled $\D W, X, Y$ and $\D Z.$ Mr Driver is approaching junction $\D A$ from Town $\D W,$ as shown, when he realises that he does not know how to get to the factory. He decides that at each road junction he will choose a road to take at random, but he will not turn around and go back along the road he has just travelled.
(a) Write down the probability that Mr Driver will choose the direct road to the factory at road junction $\D A.$ (1)
(b) Show that the probability that Mr Driver will pass through exactly two road junctions and reach the factory is $\D \frac{5}{18}.$ (3)
During the journey, if Mr Driver takes the road towards Town $\D X,$ the road towards Town $\D Y$ or the road towards Town $\D Z$ he will not arrive at the factory.
(c) Find the probability that Mr Driver will not arrive. (3)

3. [Edexcel 2012, Summer, Mathematics, Paper 01, No. 26]
A bag contains 3 red balls and 5 black balls. Two balls are to be taken at random, without replacement, from the bag.
(a) Complete the probability tree diagram.

(b) Find the probability that the two balls taken are of the same colour.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(2)

4. [Edexcel 2012, Summer, mathematic B, Paper 02, no 5]
Iftekhar travels to work each day either by bus or by train. The probability that he takes the bus is 4/5. If he takes the bus, the probability that he buys a newspaper is 3/4. If he takes the train, the probability that he buys a newspaper is 2/3.
(a) Draw a tree diagram to represent this information. (4)
(b) Calculate the probability that one particular day, Iftekhar will not buy a newspaper. (3)

5. [Edexcel 2013, winter, Mathematics, Paper 01, No. 25]
There are 100 coloured discs in a bag. Of these, 25 are brown, 40 are green and the others are neither brown nor green. A disc is to be chosen at random from the bag.
(a) Calculate the probability that the disc is either brown or green.
..................................................(2)
This disc is then returned to the bag. Two discs are now to be chosen at random from the bag without replacement.
(b) Calculate the probability that one disc will be brown and one disc will be green.
..................................................(3)

6. [Edexcel 2013, Winter, mathematic B, Paper 02, no 6]

A survey was carried out into the time it took students to travel to school on Monday. Information about the results of this survey is shown in the histogram in Figure 2. No student took more than 70 minutes to travel to school. 35 students took between 30 minutes and 40 minutes to travel to school.
(a) Calculate how many students took part in the survey. (4)
One of these students is to be chosen at random.
(b) Calculate the probability that this student took more than 30 minutes to travel to school. (2)
A similar survey was carried out on Tuesday and the results were compared with those of Monday’s survey.
On Tuesday, 8 fewer students took less than 10 minutes to travel to school. The number of students that took between 10 minutes and 30 minutes to travel to school
was the same on both Monday and Tuesday. 3 more students took between 30 minutes and 40 minutes to travel to school, 5 fewer students took more than 40 minutes to travel to school. No student took more than 70 minutes to travel to school. One of the students from Tuesday’s survey is to be chosen at random.
(c) Calculate the probability that this student took more than 30 minutes to travel to school. (3)


7. [Edexcel 2013, Summer, Mathematics B, Paper 01, No. 22]
There are only red and blue counters in a bag. When a counter is taken at random from the bag, the probability that the counter is blue is $\D \frac{2}{5}$.  Given that there are 60 counters in the bag,
(a) find the number of blue counters in the bag.
.............................................................. (2)
Some more blue counters are added to the 60 counters already in the bag. The number of extra blue counters added is $\D x.$ When a counter is now taken at random from the bag, the probability that the counter is blue is $\D \frac{1}{2}.$
(b) Find the value of $\D x.$
$\D x =$ ..............................................................(2)

8. [Edexcel 2013, Summer, Mathematics B, Paper 02, No. 8]
At Trafalgar High school 120 students took examinations in Mathematics $\D (M),$ English $\D (E)$ and Science $\D (S).$ Every student passed at least one of these subjects and $\D x$ pupils passed all three subjects. 25 students passed both Mathematics and English.
(a) Write down an expression in terms of $\D x$ for the number of students who passed both Mathematics and English but not Science. (1)
Given that
18 students passed both Mathematics and Science
17 students passed both English and Science
21 students passed Mathematics only
22 students passed English only
37 students passed Science only
(b) show all this information on Figure 4. (3)

(c) Find the value of $\D x.$ (2)
(d) Find the value of
(i) n$\D (M \cup S)$
(ii) n$\D (M \cap E \cap S' )$ (2)
A student is to be chosen at random from the 120 who took examinations in Mathematics, English and Science.
(e) Given that this student passed the Science examination, find the probability that the student also passed the English examination. (3)



9. [Edexcel 2013, Summer, Mathematics B, Paper 01R, No. 11]
A bag contains 15 red balls and 20 black balls. Balls are to be taken out of the bag at random, one at a time and not replaced. Find the probability that
(a) the first ball taken out of the bag is red,
..............................................................(1)
(b) the first two balls taken out of the bag are both red.
..............................................................(2)

10. [Edexcel 2014, winter, Mathematics B, Paper 01, No. 15]
A box contains balls of different colours. The box is opened and a ball is selected at random. The probability that the ball is white is 0.9 and the probability that the ball is black is 0.04
(a) Write down the probability that the ball is either white or black.
..............................................................(1)
(b) Find the probability that the ball is neither white nor black.
..............................................................(2)

11. [Edexcel 2014, winter, Mathematics B, Paper 02, No. 6]
On school days, Fatima goes to school by bus. The probability that it will rain on a school day is $\D \frac{2}{7}.$ When it rains, the probability that the bus will be late is $\D \frac{1}{5}.$ When it does not rain, the probability that the bus will not be late is $\D \frac{5}{6}.$
(a) Complete the probability tree diagram.

Calculate the probability that on a school day,
(b) it will be raining and the bus will be late, (2)
(c) the bus will be late. (3)

12. [Edexcel 2014, winter, Mathematics, Paper 02R, No. 7]
A sports club has 80 members. For the three activities Swimming $\D (S),$ Cycling $\D (C)$ and Running $\D (R),$
8 members take part in all three activities,
3 members do not take part in any of the three activities,
22 members take part in only Swimming,
23 members take part in Swimming and Cycling,
19 members take part in Swimming and Running,
14 members take part in Cycling and Running.
(a) Using this information place the number of members in the appropriate subsets of the Venn diagram. (3)

The number of members who take part in only Cycling is twice the number of members who take part in only Running. Let the number of members who take part in only Running be $\D x$ and, using all the given information,
(b) form an equation in $\D x.$ (1)
(c) Solve your equation to find the value of $\D x.$ (2)

13. [Edexcel 2014, Summer, mathematic B, Paper 01, no 27]
An archer shoots an arrow at a target. The probability that he will hit the target is 3/4. After the first shot, the target is moved further away from the archer. The archer shoots a second arrow at the target and the probability that he will hit the target is now 3/5.
(a) Complete the probability tree diagram.

Calculate the probability that the archer will
(b) hit the target with his first shot but miss the target with his second shot,
.......................................................(2)
(c) hit the target at least once if he takes both shots.
.......................................................(3)


14. [Edexcel 2014, Summer, mathematic B, Paper 02, no 6]
There are 159 people living in a street. The table below shows information about the number of people living in each of 30 houses in the street.
$\D\begin{array}{|c|c|}\hline
\mbox{Number (n) of people}&\mbox{Number of houses with n}\\
\mbox{living in a house}&\mbox{people living in the house}\\
\hline
1& 2\\
2& 3\\
3 &1\\
4 &4\\
5 &3\\
6 &6\\
7 &8\\
8 &2\\
9 &1\\
\hline
\end{array}$
(a) Find
(i) the modal number of people living in a house,
(ii) the median number of people living in a house,
(iii) the mean number of people living in a house.
(5)
Two houses in the street are chosen at random.
(b) Calculate the probability that 4 people live in one of the houses and 2 people live in the other of the houses. (2)
One of the people living in the street is chosen at random.
(c) Find the probability that this person lives in a house in which at least 5 people live. (2)

15. [Edexcel 2014, Summer, mathematic B, Paper 02R, no 5]
A bag contains 10 counters. Of these counters, 7 are black and 3 are white. Two of these counters are to be taken at random, without replacement, from the bag.
(a) Complete the probability tree diagram. (3)

(b) Find the probability that the two counters taken are of different colours. (3)

Answer
1.(a) $\D \frac{4}{13},\frac{7}{13}$
$\D \frac{3}{13},\frac{6}{13}$
(b) $\D \frac{3}{14}\times\frac{2}{13}$
2(a) $\D \frac{1}{3}$
(b) $\D \frac{1}{3}\times\frac{1}{3}+\frac{1}{3}\times\frac{1}{2}$
(c) $D \frac{1}{3}$
3(a) $\D 5/8,3/7,4/7$
(b) $\D \frac{13}{28}$
4 (a)%fig
(b) $\D \frac{4}{15}$
5(a) $\D \frac{13}{20}$
(b) $\D \frac{22}{99}$
6(a) 166
(b) $\D \frac{28}{83}$
(c) $\D \frac{9}{26}$
7(a) $D x=24$
(b) 12
8 (a) $\D 25-x$
(b) %fig
(c) 10
(d) (i) 98(ii) 15
(e) $\D \frac{17}{64}$
9 (a) 3/7
(b) 3/17
10 (a) 0.94
(b) 0.06
11 (a) %fig
(b) 2/35
(c) 37/210
12 (a) 3,22,15,11,6
(b) 80
(c) 5
(d) Swimming
(e)(i)39(ii) 34(iii) 8/39
13 (a) %fig
(b) 3/10
(c) 9/10
14 (a) (i) 7 (ii) 6 (iii) 5.3
(b) 24/870
(c) 132/159
15 (a) 3/10,
6/9,3/9,
7/9,2/9
(b) 7/15

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