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John keeps a weather chart. He records each day as either wet or dry. If it is dry one day, the probability that it will be dry the next day is 4/5 If it is wet one day, the probability that it will be wet the next day is 2/5 The 31st March is dry.

Calculate the probability that at least one of the days, 1st and 2nd April, will be dry.

2 Answers 2

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Case 1 - If day is dry.

Next day is dry probability $\frac45$

Next day is wet probability $\frac15$

Case 2 - If day is wet.

Next day is wet probability $\frac25$

Next day is dry probability $\frac35$

Now 31st march is dry. So we have 3 cases.

Case 1 -

If 1st april dry then provability $\frac45$ and 2nd is wet $\frac15$.

$\frac45 \times \frac15 = \frac{4}{25}$

Case 2 -

If 1st april wet then provability $\frac15$ as wet after dry and 2nd is dry $\frac35$ as dry after wet.

$\frac15 \times \frac35 = \frac{3}{25}$

Case 3 -

Both 1st april and 2nd april dry.

$\frac45 \times \frac45 = \frac{16}{25}$

Combining 3 cases we have -

$\frac{4}{25} + \frac3{25} + \frac{16}{25} = \frac{23}{25}$

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    Now ask your doubts. – 2017-01-22
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    how do you calculate the probability that at least one of the days, 1st and 2nd April, will be dry? – 2017-01-22
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    As 31st is dry. And the probability of next day dry is 4/5 as 1st april is already. Similarly 1st is dry so after dry next day is dry 4/5 as 2nd april is also dry. – 2017-01-22
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    Or in simple words dry day after dry has probability 4/5. And 31st march, 1st april and 2nd april are consecutive dry days so each has 4/5 probability. – 2017-01-22
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    Is it clear now? – 2017-01-22
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    yes, thanks very much – 2017-01-22
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    I think you are only asking question and not accepting answers even you understand. That's not good. – 2017-01-22
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    I think its one of the reason others are criticizing you. – 2017-01-22
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Say that $D_i$ means day i is dry and do something equivalent with wetness, say W. Write:

$$ P(D_{i}|D_{i-1})=4/5 \qquad P(W_{i}|W_{i-1})=2/5 $$ Now, "at least one dry day" is the complement of "no dry Days". this can only happen in 1 way, if april first and second is wet, so it will be $$ \frac{1}{5} * \frac{2}{5}=\frac{2}{25} $$ Now subtract this from1 to get your answer.