Solutions: Probability - Test 2

Q1. (b) $\frac{1}{6}$

Doublets: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6). Total 6 outcomes.
$P(\text{Doublet}) = \frac{6}{36} = \frac{1}{6}$.

Q2. (a) $\frac{3}{26}$

Black face cards: J, Q, K of Spades and Clubs. Total 6.
$P = \frac{6}{52} = \frac{3}{26}$.

Q3. (a) $\frac{1}{6}$

Sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Total 6.
$P = \frac{6}{36} = \frac{1}{6}$.

Q4. (a) Both A and R are true and R is the correct explanation of A.

Total cards = 52. Red cards = 26. $P(\text{Red}) = \frac{26}{52} = \frac{1}{2}$.

Q5. Total outcomes = 36.

(i) Sum 9: {(3,6), (4,5), (5,4), (6,3)}. Count = 4. $P = \frac{4}{36} = \frac{1}{9}$.
(ii) Sum 12: {(6,6)}. Count = 1. $P = \frac{1}{36}$.

Q6. Total cards = 52.

(i) Red King: (K-Heart, K-Diamond). Count = 2. $P = \frac{2}{52} = \frac{1}{26}$.
(ii) Queen or Jack: 4 Queens + 4 Jacks = 8. $P = \frac{8}{52} = \frac{2}{13}$.

Q7. Total outcomes = 36.

(i) 5 will not come up: Total - (5 comes up).
5 comes up: {(1,5), (2,5), (3,5), (4,5), (5,5), (6,5), (5,1), (5,2), (5,3), (5,4), (5,6)}. Count = 11.
Not 5: $36 - 11 = 25$. $P = \frac{25}{36}$.
(ii) 5 comes up at least once: Count = 11. $P = \frac{11}{36}$.

Q8. Black face cards removed (6 cards: J, Q, K of Spades/Clubs).
Remaining cards = $52 - 6 = 46$.

(i) Face card: Only red face cards remain (6). $P = \frac{6}{46} = \frac{3}{23}$.
(ii) Red card: All 26 red cards are present. $P = \frac{26}{46} = \frac{13}{23}$.
(iii) Black card: $26 - 6 = 20$. $P = \frac{20}{46} = \frac{10}{23}$.

Q9. Total outcomes = 36.

(i) Product 6: {(1,6), (2,3), (3,2), (6,1)}. Count = 4. $P = \frac{4}{36} = \frac{1}{9}$.
(ii) Product 12: {(2,6), (3,4), (4,3), (6,2)}. Count = 4. $P = \frac{4}{36} = \frac{1}{9}$.
(iii) Product 7: No integer pair (1-6) multiplies to 7. Count = 0. $P = 0$.

Q10. Removed: K, Q, J of clubs (3 cards).
Remaining cards = $52 - 3 = 49$.

(i) A heart: 13 hearts present. $P = \frac{13}{49}$.
(ii) A king: 3 kings left (Spade, Heart, Diamond). $P = \frac{3}{49}$.
(iii) A club: $13 - 3 = 10$ clubs left. $P = \frac{10}{49}$.
(iv) The '10' of hearts: 1 card. $P = \frac{1}{49}$.
(v) A face card: $12 - 3 = 9$ face cards left. $P = \frac{9}{49}$.

Q11. Total outcomes = 36.

(i) Sum is prime (2, 3, 5, 7, 11):
Sum 2: (1,1)
Sum 3: (1,2), (2,1)
Sum 5: (1,4), (2,3), (3,2), (4,1)
Sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
Sum 11: (5,6), (6,5)
Total outcomes = $1 + 2 + 4 + 6 + 2 = 15$.

(ii) Rahul wins (Prime sum): $P(\text{Rahul}) = \frac{15}{36} = \frac{5}{12}$.

(iii) Ravi wins (Composite sum):
$P(\text{Ravi}) = 1 - P(\text{Rahul}) = 1 - \frac{5}{12} = \frac{7}{12}$.