Respuesta :
Answer:
a) 1/4
b) 1/2
c) 9/64
Step-by-step explanation:
The possible outcome of five independent flip = 2^5
= 32
a) In order to determine the probability that the first three flips are the same, we have to list out the possible outcomes.
The first three flips being the same could be
HHH, TTT
= 2 ways
The last two flips could be
HH, HT, TH, TT
= 4 ways
Therefore the possible ways of obtaining the first three flips as being the same in five independent flips = 2*4
= 8 ways
HHHHH, HHHHT, HHHTH, HHHTT,
TTTHH, TTTHT, TTTTH, TTTTT
Therefore the possible ways that the first three flips are the same = 8/32
= 1/4
b) To also determine that the last three flips are the same, we will list out the possible outcomes of the last three flips
TTT, HHH
= 2 ways
The first two flips could be
HH, HT, TH, TT
= 4ways
Therefore, the possible ways to obtain the last three flips as the same in five independent flips = 2*4
= 8 ways
HHTTT, HTTTT, THTTT, TTTTT
HHHHH, HTHHH, THHHH, TTHHH
the probability that the last three flips are the same = 8/32
= 1/4
Since the probability that the first three flips are the same = 1/4
The probability that either the first three flips are the same or the last three flips are the same
= 1/4 + 1/4
= 2/4
= 1/2
c) To determine the probability that there are at least two heads among the first three flips, list out the possible outcomes.
The first three flips with at least two heads are;
HHH, THH, HHT
= 3 ways
The last two flips could be
HH, HT, TH, TT
= 4 ways
Therefore the possible ways of obtaining the first three flips with at least two heads = 3*4
= 12 ways
HHHHH, HHHHT, HHHTH, HHHTT,
THHHH, THHHT, THHTH, THHTT,
HHTHH, HHTHT, HHTTH, HHTTT
The probability that we have at least two heads among first three flips= 12/32
= 3/8
To also determine the probability that there are at least two tails among the last three flips, list out the possible outcomes.
The last three flips with at least two tails are;
TTH, THT, HTT
= 3 ways
The first two flips could be
HH, HT, TH, TT
= 4 ways
Therefore the possible ways of obtaining the last three flips with at least two tails= 3*4
= 12 ways
HHTTH, HTTTH, THTTH, TTTTH,
HHTHT, HTTHT, THTHT, TTTHT,
HHHTT, HTHTT, THHTT, TTHTT
The probability that we have at least two tails among the last three flips= 12/32
= 3/8
Therefore the probability that there are at least two heads among the first three flips and at least two tails among the last three flip = 3/8 * 3/8
= 9/64
