PROBABILITY PROBLEMS ON COINS

Problem 1:

Two coins are tossed simultaneously. Find the probability of getting

i) at least one head

ii) at most one head

iii) exactly two head

iv) exactly one head

v) no head

vi) no tail

vii) at least one tail

viii) at most one tail

ix) exactly two tails

x) exactly one tail

Solution:

Two coins are tossed = {HH, TT, HT, TH}

i) at least one head

= { HT, TH, HH}

P (getting at least one head) = 3/4

ii) at most one head

= {TT, HT, TH}

P (getting at most one head) = 3/4

iii) exactly two head

= {H H}

P (getting exactly two head) = 1/4

iv) exactly one head

= {HT, TH}

P (getting exactly one head) = 2/4

= 1/2

v) no head

= {T T}

P (getting no head) = 1/4

vi) no tail

= {HH}

P (getting no tail) = 1/4

vii) at least one tail

= {H T, T T, T H}

P (getting at least one tail) = 3/4

viii) at most one tail

= { HH, HT, TH |

P (getting at most one tail) = 2/4

= 1/2

ix) exactly two tails

= { TT }

P (getting exactly two tails) = 1/4

x) exactly one tail

= {H T, T H}

P (getting exactly one tail) = 2/4

= 1/2

Problem 2 :

A coin is tossed two times. Find the probability of getting at most one head.

Solution :

The possible outcomes, if a coin is tossed 2 times is  

S = {HH, HT, TH, TT}

Let E = Event of getting at most one head = {TT, HT, TH}

n(S) = 3

P (getting at most one head) = 3/4

Problem 3 :

A coin is tossed 3 times. List the possible outcomes. Find the probability of getting (i) all heads (ii) at least 2 heads

Solution :

The possible outcomes, if a coin is tossed 3 times is  

S = {HHH, TTT, HTT, THT, TTH, THH, HTH, HHT}

n(S) = 8

(i)           Let E₁ = Event of getting all heads = {HHH}

n(E₁) = 1

P (getting all heads) = 1/8

(ii)          Let E₂ = Event of getting atleast 2 heads = {HHT, HTH, THH, HHH}

n(E₁) = 4

n(E₁) = 4

P (getting atleast 2 heads) = 4/8

= 1/2

Problem 4 :

Sushma tosses a coin 3 times and gets tail each time. Do you think that the outcomes of next toss will be a tail? Give reasons.

Solution :

The outcome of next toss may or may not be tail, because on tossing a coin, we get head or tail so both are equally likely events.

Problem 5 :

If I toss a coin 3 times and get head each time, should I expect a tail to have a higher chance in the 4th toss? Give reason in support of your answer.

Solution :

False, because of the outcomes ‘head’ and ‘tail’ are equally, likely. So, every time the probability of getting head or tail is 1/2.

Problem 6:

Three coins are tossed simultaneously. What is the probability of getting

i) exactly two heads

ii) at least two heads

iii) at most two heads

Solution :

Three coins are tossed simultaneously.

When three coins are tossed then the outcome will be

{TTT, THT, TTH, THH, HTT, HHT, HTH, HHH}

Total number of outcome = 8

i) exactly two heads

Favorable outcome= {(THH), (HHT), (HTH)}

P (getting exactly two heads) = 3/8             

iii) at least two heads

Favorable outcome= {HHT, HTH, HHH, THH}

P (getting at least two heads) = 4/8

= 1/2

iii) at most two heads

Favorable outcome= {HHT, HTH, TTT, THH, TTH, THT, HTT}

P (getting at most two heads) = 7/8