Question

1 point
A fair coin is tossed three times and a T (for tails) or H (for heads) is recorded, giving us a list of length 3. Let X be the random variable which is zero if no T has another T adjacent to it, and is one othetwise. Let Y denote the random variable that counts the number of T's in the three tosses. Find P(X=1, Y=2).
A) 1/8
B) 2/8
C) 5/8
D) 7/8

Answer 1

Answer on Question #81422 – Math – Statistics and Probability

Question

A fair coin is tossed three times and a $T$ (for tails) or $H$ (for heads) is recorded, giving us a list of length 3. Let $X$ be the random variable which is zero if no $T$ has another $T$ adjacent to it, and is one otherwise. Let $Y$ denote the random variable that counts the number of $T$'s in the three tosses. Find $P(X = 1, Y = 2)$.

A) 1/8

B) 2/8

C) 5/8

D) 7/8

Solution

When a fair coin is tossed 3 times, the outcomes are

$$\{TTT, TTH, THT, THH, HTT, HTH, HHT, HHH\}$$

$X = 1$ implies there is a $T$ adjacent to a $T$ and $Y = 2$ implies there are two $T$

There are two outcomes satisfying this: $TTH, HTT$

Hence,

$$P(X = 1, Y = 2) = \frac{2}{8}.$$

Answer: B) 2/8.

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