Question 1: Two customers Shyam and Ekta are visiting a particular shop in the same week (Tuesday to Saturday). Each is equally likely to visit the shop on any day as on another day. What is the probability that both will visit the shop on (i) the same day? (ii) consecutive days? (iii) different days?

**Solution:** The total number of days is 5 and hence both of them can reach the shop in 5 ways.

Hence, total number of outcomes `= 5 xx 5 = 25`

They can reach on the same day in 5 ways, i.e. (Tue Tue), (Wed Wed), (Thur Thur), (Fri Fri) and (Sat Sat)

P(Reaching on same day) `=5/25=1/5`

They can reach on consecutive days in following 8 ways: (tue wed), (wed, tue), (wed thur), (thur wed), (thur fri), (fri thu), (fri sat), (sat fri)

P(Reaching on consecutive days) `=8/25`

Since P (reaching on same days) `= 1/5`

Hence, P(reaching on different days) `=1-1/5=4/5`

Question 2: A die is numbered in such a way that its faces show the numbers 1, 2, 2, 3, 3, 6. It is thrown two times and the total score in two throws is noted. Complete the following table which gives a few values of the total score on the two throws:

What is the probability that total score is (i) even? (ii) 6? (iii) at least 6?

**Solution:** Following table shows the sample space:

+ | 1 | 2 | 2 | 3 | 3 | 6 |

1 | 2 | 3 | 3 | 4 | 4 | 7 |

2 | 3 | 4 | 4 | 5 | 5 | 8 |

2 | 3 | 4 | 4 | 5 | 5 | 8 |

3 | 4 | 5 | 5 | 6 | 6 | 9 |

3 | 4 | 5 | 5 | 6 | 6 | 9 |

6 | 7 | 8 | 8 | 9 | 9 | 12 |

Total number of outcomes = 36

Number of even score = 18

Hence, P (even score) `=18/36=1/2`

Number of times 6 comes = 4

Hence, P (score of 6) `=4/36=1/9`

Number of times score is 6 or more = 15

Hence, P (score at least 6) `=15/36`

Question 3: A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, determine the number of blue balls in the bag.

**Solution:** Let us assume number of blue balls `= x`

Then, total number of balls `= x + 5`

Then;

P(Red) `=(5)/(5+x)`

P(Blue) `=(x)/(5+x)`

As per question;

`(x)/(5+x)=2xx(5)/(5+x)`

Or, `x=2xx5=10`

Number of blue balls = 10

Question 4: A box contains 12 balls out of which x are black. If one ball is drawn at random from the box, what is the probability that it will be a black ball? If 6 more black balls are put in the box, the probability of drawing a black ball is now double of what is was before. Find x.

**Solution:** Total number of outcomes = 12

`P(x)=x/12`

When 6 more black balls are put in the box, total number of balls = 12 + 6 = 18

Number of black balls = 6 + x

P(Black) `=(x+6)/(18)`

As per question;

`(x+6)/(18)=2xx(x)/(12)`

Or, `x+6=3x`

Or, `2x=6`

Or, `x=3`

Question 5: A jar contains 24 marbles, some are green and others are blue. If a marble is drawn at random from the jar, the probability that it is green is 2/3. Find the number of blue balls in the jar.

**Solution:** Total number of outcomes = 24

P (Green) `= 2/3`

If number of green balls is G then;

`G/24=2/3`

Or, `G=(24xx2)/(3)=16`

Hence, number of blue balls = 24 – 16 = 8

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