Respuesta :
Answer:
(a) S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
(b) [tex]\frac{1}{8}[/tex]
(c) [tex]\binom{3}{2} (\frac{1}{2})^3 = \frac{3}{8}[/tex]
(d) [tex](\frac{1}{2})^3 + \binom{3}{1} (\frac{1}{2})^3 = \frac{1}{8} + \frac{3}{8} = \frac{1}{2}[/tex]
(e) [tex]1 - \frac{1}{2} = \frac{1}{2}[/tex]
(f) Yes, because [tex]P(A \cap B) = 0[/tex]. Therefore [tex]P(A \cup B) = P(A) + P(B) = \frac{3}{8} + \frac{1}{2} = \frac{7}{8}[/tex]
(g) No, because [tex]P(A \cap C) = \frac{3}{8}[/tex]. Therefore [tex]P(A \cup C) = P(A) + P(C) - P(A \cap C) = \frac{3}{8} + \frac{1}{2} - \frac{3}{8} = \frac{1}{2}[/tex]
A) Sample space, S = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT },
B) The probability associated with each outcome is 1/8
C) The probability of the event of exactly 2 heads is 3/8
D) The probability of the event of at most 1 head is 3/8
E) The probability of the event of at least 2 heads is 1/2
F) Yes, the events A and B mutually exclusive events and P(A or B) is 3/4.
G) No, the events A and C are not mutually exclusive events.
P(A or C) is 1/2
What is probability?
Probability deals with the occurrence of a random event. The chance that a given event will occur. It is the measure of the likelihood of an event to occur.The value is expressed from zero to one.
For the given situation,
Three coins are dropped on a table.
A) All possible outcomes in the sample space are
[tex]S =[{ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }][/tex]
⇒ n(s) = 8
B) The probability associated with each outcome
Total number of samples is 8. So the probability associated with each outcome is [tex]P(e)=\frac{1}{8}[/tex].
C) Let A be the event “exactly 2 heads”.
[tex]A = [{HHT, HTH, THH}][/tex]
⇒ [tex]n(A)=3[/tex]
Thus probability of the event of exactly 2 heads is
⇒ [tex]P(A)=\frac{n(A)}{n(S)}[/tex]
⇒ [tex]P(A)=\frac{3}{8}[/tex]
D) Let B be the event “at most 1 head”
[tex]B=[{HTT, THT,TTH}][/tex]
⇒ [tex]n(B)=3[/tex]
Thus probability of the event of at most 1 head is
⇒ [tex]P(B)=\frac{n(B)}{n(S)}[/tex]
⇒ [tex]P(B)=\frac{3}{8}[/tex]
E) Let C be the event “at least 2 heads".
C be the event “at least 2 heads is
⇒ [tex]C={[HHH, HHT, HTH, THH]}[/tex]
Thus probability of the event of at least 2 heads is
[tex]P(C)=\frac{4}{8}[/tex]
⇒ [tex]P(C)=\frac{1}{2}[/tex]
F) The events A and B mutually exclusive
The event A and B has no element in common
⇒ [tex]A \cap B=0[/tex]
So the events A and B are mutually exclusive events.
Then, [tex]P(A or B) = P(A)+P(B)[/tex]
⇒ [tex]P(A or B) = \frac{3}{8} +\frac{3}{8}[/tex]
⇒ [tex]P(A or B) = \frac{6}{8}[/tex]
⇒ [tex]P(A or B) = \frac{3}{4}[/tex]
G) The events A and C mutually exclusive
The event A and C has 3 element in common.
⇒ [tex]A \cap B=3[/tex]
So the events A and C are not mutually exclusive events.
Then, [tex]P(A or C) = P(A)+P(C)-P (A \cap C)[/tex]
⇒ [tex]P(A or C) = \frac{3}{8} +\frac{1}{2} -\frac{3}{8}[/tex]
⇒ [tex]P(A or C) = \frac{3+4-3}{8}[/tex]
⇒ [tex]P(A or C) = \frac{4}{8}[/tex]
⇒ [tex]P(A or C) = \frac{1}{2}[/tex]
Hence we can conclude that
A) Sample space, S = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT },
B) The probability associated with each outcome is 1/8
C) The probability of the event of exactly 2 heads is 3/8
D) The probability of the event of at most 1 head is 3/8
E) The probability of the event of at least 2 heads is 1/2
F) Yes, the events A and B mutually exclusive events and P(A or B) is 3/4.
G) No, the events A and C are not mutually exclusive events.
Learn more about probability here
https://brainly.com/question/14818004
#SPJ2
