Course: Mathematics-I (CDA101), Topic- Probability
Tutorial Sheet #2
1. Prove the following identities:
(a) nk = n
n−k
.
(b) nk + n n+1
k+1
= k+1
.
(c) Pnk=0 n
P n
k
=2 .
(d) P nk=0 n
k
(−1) k
= 0.
(e) P nk=1 n
k
k = n2n−1 .
(f) nk=0 n
k
k(−1)k = 0.
2. If two events A and B can occur and Pr(A) is not zero and Pr(B) is not zero, what
combinations of independent (I), not independent (NI), mutually exclusive (M),
and not mutually exclusive (NM) are permissible? In other words, which of the four
combinations (I, M), (NI, M), (I, NM), and (NI, NM) are permissible? Construct
an example for those combinations that are permissible.
3. Cards are drawn from a standard 52-card deck until the third club is drawn. After
each card is drawn, it is put back in the deck and the cards are reshuffled so that
each card drawn is independent of all others.
(a) Find the probability that the 3rd club is drawn on the 8th selection.
(b) Find the probability that at least 8 cards are drawn before the 3rd club appears.
(c) Repeat parts (a) and (b) if the cards are drawn without replacement. That is,
after each card is drawn, the card is set aside and not replaced in the deck.
4. I deal myself 3 cards for a standard 52-card deck. Find the probabilities of each of
the following events:
(a) 2 of a kind (e.g., 2 fives or 2 kings),
(b) 3 of a kind,
(c) 3 of the same suit (a.k.a. a flush, e.g., 3 hearts or 3 clubs),
(d) 3 cards in consecutive order (a.k.a. a straight, e.g., 2-3-4 or 10-J-Q).
5. I deal myself 5 cards for a standard 52-card deck. Find the probabilities of each of
the following events:
(a) 2 of a kind,
(b) 3 of a kind,
(c) 2 pair (e.g., 2 eights and 2 queens),
(d) a flush (5 cards all of the same suit),
(e) a full house (3 of one kind and 2 of another kind),
(f) a straight (5 cards in consecutive order).
6. I deal myself 13 cards for a standard 52-card deck. Find the probabilities of each of
the following events:
(a) exactly one heart appears in my hand (of 13 cards);
(b) at least 7 cards from a single suit appear in my hand;
(c) my hand is void (0 cards) of at least one suit.
1
Tutorial Sheet #2
1. Prove the following identities:
(a) nk = n
n−k
.
(b) nk + n n+1
k+1
= k+1
.
(c) Pnk=0 n
P n
k
=2 .
(d) P nk=0 n
k
(−1) k
= 0.
(e) P nk=1 n
k
k = n2n−1 .
(f) nk=0 n
k
k(−1)k = 0.
2. If two events A and B can occur and Pr(A) is not zero and Pr(B) is not zero, what
combinations of independent (I), not independent (NI), mutually exclusive (M),
and not mutually exclusive (NM) are permissible? In other words, which of the four
combinations (I, M), (NI, M), (I, NM), and (NI, NM) are permissible? Construct
an example for those combinations that are permissible.
3. Cards are drawn from a standard 52-card deck until the third club is drawn. After
each card is drawn, it is put back in the deck and the cards are reshuffled so that
each card drawn is independent of all others.
(a) Find the probability that the 3rd club is drawn on the 8th selection.
(b) Find the probability that at least 8 cards are drawn before the 3rd club appears.
(c) Repeat parts (a) and (b) if the cards are drawn without replacement. That is,
after each card is drawn, the card is set aside and not replaced in the deck.
4. I deal myself 3 cards for a standard 52-card deck. Find the probabilities of each of
the following events:
(a) 2 of a kind (e.g., 2 fives or 2 kings),
(b) 3 of a kind,
(c) 3 of the same suit (a.k.a. a flush, e.g., 3 hearts or 3 clubs),
(d) 3 cards in consecutive order (a.k.a. a straight, e.g., 2-3-4 or 10-J-Q).
5. I deal myself 5 cards for a standard 52-card deck. Find the probabilities of each of
the following events:
(a) 2 of a kind,
(b) 3 of a kind,
(c) 2 pair (e.g., 2 eights and 2 queens),
(d) a flush (5 cards all of the same suit),
(e) a full house (3 of one kind and 2 of another kind),
(f) a straight (5 cards in consecutive order).
6. I deal myself 13 cards for a standard 52-card deck. Find the probabilities of each of
the following events:
(a) exactly one heart appears in my hand (of 13 cards);
(b) at least 7 cards from a single suit appear in my hand;
(c) my hand is void (0 cards) of at least one suit.
1
