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adell [148]
2 years ago
8

A fair coin is tossed three times and the events A, B, and C are defined as follows: A: \{ At least one head is observed \} B: \

{ At least two heads are observed \} C: \{ The number of heads observed is odd \} Find the following probabilities by summing the probabilities of the appropriate sample points (note that 0 is an even number):
(a) P(A) =
(b) P(A \textrm{ or } B) =
(c) P((\textrm{not }A) \textrm{ or } B \textrm{ or } (\textrm{not }C)) =
Mathematics
1 answer:
Yanka [14]2 years ago
4 0

Answer:

a) P(A)=0.875

b) \text{P(A or B)}=0.875

c) \text{P((not A)  or B  or (not C))}=0.625

Step-by-step explanation:

Given : A fair coin is tossed three times and the events A, B, and C are defined as follows: A: At least one head is observed, B: At least two heads are observed, C: The number of heads observed is odd.

To find : The following probabilities by summing the probabilities of the appropriate sample points ?

Solution :

The sample space is

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

n(S)=8

A: At least one head is observed

i.e. A={HHH,HHT,HTT,HTH,TTH,TTH,THH,THT}

n(A)=7

B: At least two heads are observed

i.e. B={HHH,HTT,TTH,THT}

n(B)=4

C: The number of heads observed is odd.

i.e. C={HHH,HTT,THT,TTH}

n(c)=4

a) Probability of A, P(A)

P(A)=\frac{n(A)}{n(S)}

P(A)=\frac{7}{8}

P(A)=0.875

b) P(A or B)

Using formula,

\text{P(A or B)}=P(A)+P(B)-\text{P(A and B)}

\text{P(A or B)}=\frac{n(A)}{n(S)}+\frac{n(B)}{n(S)}-\frac{\text{n(A and B)}}{n(S)}

\text{P(A or B)}=\frac{7}{8}+\frac{4}{8}-\frac{4}{8}

\text{P(A or B)}=\frac{7}{8}

\text{P(A or B)}=0.875

(c) P((not A)  or B  or (not C))

A={HHH,HHT,HTT,HTH,TTH,TTH,THH,THT}

not A = {TTT} = 1

B={HHH,HTT,TTH,THT}

C={HHH,HTT,THT,TTH}

not C = {HHT,HTH,THH,TTT} = 4

So, not A or B or not C = {HHH,HHT,HTH,THH,TTT}=5

\text{P((not A)  or B  or (not C))}=\frac{5}{8}

\text{P((not A)  or B  or (not C))}=0.625

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