Answer:
a) ![P(A)=0.875](https://tex.z-dn.net/?f=P%28A%29%3D0.875)
b) ![\text{P(A or B)}=0.875](https://tex.z-dn.net/?f=%5Ctext%7BP%28A%20or%20B%29%7D%3D0.875)
c) ![\text{P((not A) or B or (not C))}=0.625](https://tex.z-dn.net/?f=%5Ctext%7BP%28%28not%20A%29%20%20or%20B%20%20or%20%28not%20C%29%29%7D%3D0.625)
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)}](https://tex.z-dn.net/?f=P%28A%29%3D%5Cfrac%7Bn%28A%29%7D%7Bn%28S%29%7D)
![P(A)=\frac{7}{8}](https://tex.z-dn.net/?f=P%28A%29%3D%5Cfrac%7B7%7D%7B8%7D)
![P(A)=0.875](https://tex.z-dn.net/?f=P%28A%29%3D0.875)
b) P(A or B)
Using formula,
![\text{P(A or B)}=P(A)+P(B)-\text{P(A and B)}](https://tex.z-dn.net/?f=%5Ctext%7BP%28A%20or%20B%29%7D%3DP%28A%29%2BP%28B%29-%5Ctext%7BP%28A%20and%20B%29%7D)
![\text{P(A or B)}=\frac{n(A)}{n(S)}+\frac{n(B)}{n(S)}-\frac{\text{n(A and B)}}{n(S)}](https://tex.z-dn.net/?f=%5Ctext%7BP%28A%20or%20B%29%7D%3D%5Cfrac%7Bn%28A%29%7D%7Bn%28S%29%7D%2B%5Cfrac%7Bn%28B%29%7D%7Bn%28S%29%7D-%5Cfrac%7B%5Ctext%7Bn%28A%20and%20B%29%7D%7D%7Bn%28S%29%7D)
![\text{P(A or B)}=\frac{7}{8}+\frac{4}{8}-\frac{4}{8}](https://tex.z-dn.net/?f=%5Ctext%7BP%28A%20or%20B%29%7D%3D%5Cfrac%7B7%7D%7B8%7D%2B%5Cfrac%7B4%7D%7B8%7D-%5Cfrac%7B4%7D%7B8%7D)
![\text{P(A or B)}=\frac{7}{8}](https://tex.z-dn.net/?f=%5Ctext%7BP%28A%20or%20B%29%7D%3D%5Cfrac%7B7%7D%7B8%7D)
![\text{P(A or B)}=0.875](https://tex.z-dn.net/?f=%5Ctext%7BP%28A%20or%20B%29%7D%3D0.875)
(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}](https://tex.z-dn.net/?f=%5Ctext%7BP%28%28not%20A%29%20%20or%20B%20%20or%20%28not%20C%29%29%7D%3D%5Cfrac%7B5%7D%7B8%7D)
![\text{P((not A) or B or (not C))}=0.625](https://tex.z-dn.net/?f=%5Ctext%7BP%28%28not%20A%29%20%20or%20B%20%20or%20%28not%20C%29%29%7D%3D0.625)