
What is the probability that heads never occurs doubly in a rowing ?
Your proposed answer of $ 13/32 $ is decline.
If there are four or five heads in the succession of five coin tosses, at least two heads must be straight .
If there are three heads in the succession of five coin tosses, the lone possibility is that the sequence is HTHTH .
There are $ \binom { 5 } { 2 } = 10 $ sequences of five coin tosses with precisely two heads, of which four have straight heads ( since the first of these back-to-back heads must appear in one of the first four positions ). Hence, there are $ 10 – 4 = 6 $ sequences of five coin tosses with precisely two heads in which no two heads are consecutive .
In each of the five sequences of mint tosses in which precisely one pass appears, no two heads are consecutive .
In the entirely sequence of five coin tosses in which no heads appear, no two heads are straight.
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Hence, the number of sequences of five coin tosses in which no two heads are consecutive is $ 0 + 0 + 1 + 6 + 5 + 1 = 13 $, as you found .
What is the probability that neither heads nor tail occurs doubly in a row ?
Your proposed answer of $ 1/16 $ is chastise since there are only two favorable cases : HTHTH and THTHT, which gives the probability $ \frac { 2 } { 32 } = \frac { 1 } { 16 } $ .
What is the probability that both heads and tails occur at least doubly in a row ?
Your proposed answer of $ 15/16 $ is faulty.
Since $ 1 – \frac { 1 } { 16 } = \frac { 15 } { 16 } $, your answer suggests you mistakenly believed that the negation of the affirmation that neither heads nor chase occurs twice in a row is that both heads and tails occur at least twice in a course. The negation of the statement that neither heads nor tail occurs doubly in a row is that at least two heads or at least two tails are back-to-back. For example, the sequences HHTHT and TTTTH both violate the restriction that neither heads nor tails occur doubly in a row without satisfying the stronger prerequisite that both heads and tails occur at least doubly in a row .
If both heads and tails occur at least twice in a course, then there are four possibilities :
- there is a block of three consecutive heads and a block of two consecutive tails
- there is a block of three consecutive tails and a block of two consecutive heads
- there is a block of two consecutive heads and a single head that are separated by a block of two consecutive tails
- there is a block of two consecutive tails and a single tail that are separated by a block of two consecutive heads
A auction block of three back-to-back heads and a barricade of two consecutive tails can occur in two ways, HHHTT and TTHHH. By symmetry, a block of three straight tails and two straight heads can occur in two ways. A block of two consecutive heads and a individual head that are separated by a block of two consecutive tails can occur in two ways, HHTTH and HTTHH. By symmetry, a barricade of two consecutive tails and a unmarried fag end that are separated by a block of two straight heads can occur in two ways. Hence, there are $ 2 + 2 + 2 + 2 = 8 $ friendly cases, giving a probability of $ \frac { 8 } { 32 } = \frac { 1 } { 4 } $ .