Consider two coins, CH and CT . Suppose that coin CH has probability 0.6 of coming up heads and that coin CT has probability 0.3 of coming up heads. On each day, we will select one coin to flip and register the outcome. On the first day (day one), we will pick a coin at random (uniformly). On subsequent days, we will flip coin CH if the outcome the previous day was heads, and we will flip coin CT if the previous outcome was tails. Determine the probability that the coin flipped on the third day after the initial flip (i.e., on day four) is coin CH. State any independence assumptions you use. Explain.

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Ritz01 Ritz01 · Answer 1 · 2020-10-05T07:58:08+02:00

Answer:

0.4305

Step-by-step explanation:

Let [tex]P(H_n)[/tex] and [tex]P(T_n)[/tex] be the probability of coming up head and tail respectively on day [tex]n[/tex].

On day four, the coin CH will be salected iff there in head as outcome on the previous day, i.e. on the day three.

So, actually the probability of salection of coin CH on day four is probability of coming up head on day three, [tex]P(H_3)[/tex].

On day 1, n=1:

As one coin is salected randomly from two coin, so, the probability of salection of one coin is 1/2=0.5.

[tex]P(H_1)[/tex] is the probability of salection of coin CH and coming up head or salection of coin CT and coming up head when tossed.

So, [tex]P(H_1)=0.5\times 0.6 + 0.5\times0.3=0.45[/tex].

As there is only two outcome on any day, head or tail, so

[tex]P(T_1)=1-P(H_1)=1-0.45=0.55[/tex]

On day two, n=2:

The probability of salection of any coin depends on the outcpme of previous day, i.e

Probability of salection of coin CH and CT on day two are

[tex]P(H_1)=0.45[/tex] and [tex]P(T_1)=0.55[/tex] respectively.

So, [tex]P(H_2)=0.45\times 0.6 + 0.55\times0.3=0.435[/tex].

As there is only two outcome on any day, head or tail, so

[tex]P(T_2)=1-P(H_2)=1-0.435=0.565[/tex]

Similarly, on day three, n=3:

The required probability is [tex]P(H_3)[/tex],

[tex]P(H_3)=P(H_2)\times0.6 + P(T_2)\times 0.3[/tex]

[tex]\Rightarrow P(H_3)=0.435\times0.6 + 0.565\times 0.3=0.4305[/tex]

Hence, the salection od coin CH on day four is 0.4305.