Each day, a weather forecaster predicts whether or not it will rain. for 80% of rainy days, she correctly

Comments (2) on “Each day, a weather forecaster predicts whether or not it will rain. for 80% of rainy days, she correctly”

1. Our required proportion of the forecasts are correct is 89%.

   Step-by-step explanation:

   Since we have given that

   Percentage of rainy days she correctly predicts that it will rain = 80%

   Percentage of non rainy days, she correctly predicts that it will not rain = 90%

   Percentage of rainy days = 10%

   Percentage of non rainy  days = 90%

   So, Probability of forecasts are correct is given by

   P(rainy day).P(correctly predicted) + P(non rainy day) .P(correctly predicted)

   [tex]0.1\times 0.8+0.9\times 0.9\\\\=0.08+0.81\\\\=0.89\\\\=89\%[/tex]

   Hence, our required proportion of the forecasts are correct is 89%.

   Reply

2. Proportion of  correct forecast for first forecaster = 0.89 i.e. 89/100

   For second forecaster proportion of  correct forecast = 0.9 i.e. 90/100

   Step-by-step explanation:

   Consider,

   Events of rainy days = R₁

   Events of non-rainy days = R₂

   Events of correct forecast = C

   A) for first forecaster:

   correct forecast for rainy days = 80%

                                                       P(C|R₁) = 0.8

   correct forecast for non-rainy days = 90%

                                                         P(C|R₂) = 0.9

   %age rainy days = 10%

                                        P(R₁) = 0.1

   %age of non-rainy days = 90%

                                       P(R₂) = 0.9

   Using Baye's formula of conditional probability,

   proportion of correct forecast = P(C) = P(C|R₁)*P(R₁) + P(C|R₂) *P(R₂)

                                                                = (0.8)(0.1) + (.9)(0.9)

                                                                 = 0.89

   i.e. proportion of correct forecast for first forecaster = 89/100

   B) for second forecaster:

   forecast for non-rainy days = 100%

                                                         P(C|R₂) = 0.9

   forecast for non-rainy days = 0%

                                                         P(C|R₁) = 0

   Using Baye's formula of conditional probability,

   proportion of correct forecast = P(C) = P(C|R₁)*P(R₁) + P(C|R₂) *P(R₂)

                                                                = (0)(0.1) + (1)(0.9)

                                                                 = 0.90

   i.e. proportion of correct forecast for first forecaster = 90/100

   Reply