If it rains on a given day the probability that it will rain the next day is 0.65. If it does not rain on a given day the probability that it will rain the next day is 0.3. It rained on Monday. (a) Complete the tree diagram. Tuesday Rain No Rain Wednesday Rain No Rain (b) Find the probability that it rains on Wednesday.

GCSE OCR Maths Tree diagrams: If it rains on a given day the p

If it rains on a given day the probability that it will rain the next day is 0.65 - OCR - GCSE Maths - Question 16 - 2023 - Paper 4

Question 16

If it rains on a given day the probability that it will rain the next day is 0.65. If it does not rain on a given day the probability that it will rain the next day ... show full transcript

Worked Solution & Example Answer:If it rains on a given day the probability that it will rain the next day is 0.65 - OCR - GCSE Maths - Question 16 - 2023 - Paper 4

Step 1

Complete the tree diagram.

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Answer

The tree diagram for the probabilities can be filled as follows:

On Tuesday:
- If it rains: 0.65 (probability of rain the next day)
- If it does not rain: 0.35 (1 - 0.65)
On Wednesday (depending on Tuesday):
- If it rained on Tuesday:
  - Rain: 0.65
  - No Rain: 0.35
- If it did not rain on Tuesday:
  - Rain: 0.3
  - No Rain: 0.7

Thus, the completed tree diagram would look as follows:

                     Tuesday
                    /       \
                  Rain      No Rain
                 (0.65)     (0.35)
                  /   \       /   \
               Rain  No Rain Rain   No Rain
              (0.65) (0.35)  (0.3)  (0.7)

Step 2

Find the probability that it rains on Wednesday.

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Answer

To find the total probability that it rains on Wednesday, we use the Law of Total Probability:

Let:

P(Rain on Wednesday|Rained on Tuesday) = 0.65
P(Rain on Wednesday|No Rain on Tuesday) = 0.3
P(Rained on Tuesday) = 0.65
P(No Rain on Tuesday) = 0.35

Now, we can calculate:

$P(Rain ext{ on Wednesday}) = P(Rain ext{ on Wednesday|Rained on Tuesday}) imes P(Rained ext{ on Tuesday}) + P(Rain ext{ on Wednesday|No Rain on Tuesday}) imes P(No Rain ext{ on Tuesday})$

Substituting the probabilities:

$P(Rain ext{ on Wednesday}) = (0.65 imes 0.65) + (0.3 imes 0.35)$

This simplifies to:

$P(Rain ext{ on Wednesday}) = 0.4225 + 0.105 = 0.5275$

Therefore, the probability that it rains on Wednesday is 0.5275 or 52.75%.

If it rains on a given day the probability that it will rain the next day is 0.65 - OCR - GCSE Maths - Question 16 - 2023 - Paper 4

Worked Solution & Example Answer:If it rains on a given day the probability that it will rain the next day is 0.65 - OCR - GCSE Maths - Question 16 - 2023 - Paper 4

Complete the tree diagram.

Find the probability that it rains on Wednesday.

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