1.Algebra Booster

Probability 8.17
23. Find the probability that three persons have same date
and month for their birthday.
24. A natural number x is chosen at random from {0, 1, 2,
…, 9}. Find the probability of x for which the equation
3 x = 2x 2 + 1 has a solution.
25. Three integers are chosen at random from the first 20
integers. Find the probability that their product is even.
Integer Type Questions
1. A and B toss a coin each simultaneously 50 times. If
the probability that both of them will not get tail at the
50
Ê pˆ
same toss is
Á
Ë q
˜
, where p and q are relatively prime,
¯
find (p + q + 2).
2. A natural number x is chosen at random from first 10
natural numbers. If the probability of x satisfies the in-
Êaˆ
equation x 2 – 13x Á ˜ where a and b are relatively
prime, find the value of (b – a + 2).
Ëb¯ 3. A man is known to speak the truth 3 out of 4 times.
He throws a die and reports that it is three. If the probability
that it is not a three is Á ˜ where m and n are
Êmˆ
Ë n ¯, Ê n ˆ
natural numbers, find the value of Á
Ëm
+ 1˜
¯.
4. 3 mangoes and 3 apples are in a box. If fruits are chosen
at random such that the probability of one mango
Êaˆ
and one apple is Á ˜ , where a and b are prime numbers,
find the value of (a + b).
Ëb¯ 5. A man alternately tosses a coin and throws a die. If the
probability of getting a head on the coin before he gets
six on the die is p, find (7p + 1)
6. The probability of getting a sum of 12 in four throws
3
1
of an ordinary dice is
Êmˆ Á ˜ ¥ , where m and n are
Ë n ¯ 6
relatively prime numbers, find the value of (n – m + 2).
7. Two numbers a and b are chosen at random from the
set of integers {1, 2, 3, …, 15}. If the probability that
the equation 2a – 3b = 0 is satisfied is p, find the value
of (84p + 2).
8. Two numbers x and y are chosen at random from
{1, 2, 3, …, 9}. If the probability of (x 3 + y 3 ) is divisible
Ê pˆ
by 3 is
Á
Ë q
˜ , where p and q are positive integers, find
¯
(p + q + pq).
9. A box contains 24 identical balls out of which 8 are
black and 16 are white. The balls are drawn at random
from the box one at a time with replacement. If the
probability that a white ball is drawn for the 4th time on
3
the 7th draw is Êaˆ 40
Á ˜ ¥ , where a and b are relatively
Ëb¯
81
prime numbers, find the value of (a + b + 1).
10. Let X be a universal set and A and B be two subsets of
it. If the probability of selecting 2 subsets A and B such
that B = A is 1/127, find the number of elements in the
set X.
11. A box contains 3 white, 2 black and 4 red balls. Four
balls are drawn at random with replacement. If the
probability that the sample contains only one white ball
4
is
Êaˆ
2 ¥ Á ˜ , where a and b are relatively prime numbers,
find (a + b + 1).
Ëb¯ 12. If the probability that a randomly chosen year of
the 22nd Century will have 53 Sundays is Êaˆ 1
Á ˜ ¥ ,
Ëb¯
7
where a and b are relatively prime, find the value of
(a + b – 3).
13. Two dice are thrown simultaneously. If the probability
Êmˆ
that the sum of two numbers will be 5 before 7 is Á
Ë
˜
n ¯,
where m and n are prime numbers, find the value of
(m + n).
Comprehensive Link Passage
Passage 1
There are four boxes A 1
, A 2
, A 3
and A 4
. Box A 1
has i cards and
on each card a number is printed, the numbers are from 0 to
i. A box is selected randomly, the probability of selection of
box A i
is 1/10 and then a card is drawn. Let E i
denotes the
event that a card with number i is drawn.
(i) P(E i
) is equal to
(a) 1/5 (b) 1/10 (c) 2/5 (d) 1/4
(ii) P(A 3
/E 2
) is equal to
(a) 1/4 (b) 1/3 (c) 1/2 (d) 2/3
(iii) Expectation of the number on the card is
(a) 2 (b) 2.5 (c) 3 (d) 3.5
Passage 2
A biased coin shows head with a probability 3/4 and tails
with a probability 1/4. Let P n
denotes the probability that no
three or more heads appear consecutively in n throws of the
coins.
On the basis of the above information, answer the following
questions.
(i) If P n
= aP n–3
+ bP n–2
+ gP n–3
, the value of
64(a + b + g) is
(a) 37 (b) 23 (c) 121 (d) 119.
(ii) The value of P 4
is
(a) 121/256 (b) 23/64 (c) 37/64 (d) 119/256
(iii) If the coin is tossed 4 times and let A be the event that
three or more head occurs in four tosses and B is the