Question

Iз 18 стрiльцiв п’ятеро влучають у мiшень з iмовiрнiстю 0.8, семеро – з iмовiрнiстю 0.7, четверо – з iмовiрнiстю 0.6 i двоє – з iмовiрнiстю 0.5. Навмання обраний стрiлець вистрiлив, але у мiшень не влучив. До якої групи вiн найiмовiрнiше належить?

Answer 1

Відповідь:

Пояснення:

Нехай подія Н1 полягає в тому, що стрілець, який влучає у мiшень з iмовiрнiстю 0.8. Н2-з iмовiрнiстю 0.7; Н3 - з iмовiрнiстю 0.6; Н4- з iмовiрнiстю 0.5

Подія А - стрілець у мiшень не влучив.

Р(Н1)=5/18. Р(А/Н1)=1-0.8=0.2

Р(Н2)=7/18. Р(А/Н2)=1-0.7=0.3

Р(Н3)=4/18. Р(А/Н3)=1-0.6=0.4

Р(Н4)=2/18. Р(А/Н4)=1-0.5=0.5

Підрахуємо Р(А)=Р(Н1)×Р(А/Н1)+Р(Н2)×Р(А/Н2)+ Р(Н3)×Р(А/Н3)+Р(Н4)×Р(А/Н4)= 1/18×(5×0.2+7×0.3+4×0.4+2×0.5)=5.7/18=0.3167

Р(Н1/А)=Р(Н1)Р(А/Н1)/Р(А)=5/18×0.2/0.3167=0.1754

Р(Н2/А)=Р(Н2)Р(А/Н2)/Р(А)=7/18×0.3/0.3167=0.3684

Р(Н3/А)=Р(Н3)Р(А/Н3)/Р(А)=4/18×0.4/0.3167=0.2807

Р(Н4/А)=Р(Н4)Р(А/Н4)/Р(А)=2/18×0.5/0.3167=0.1754

Найбільша ймовірність, що стрілець належав до другої групи Н2

Answer 2

Problem Analysis

We are given information about the probability of different groups of shooters hitting the target. We need to determine which group the shooter is most likely to belong to, given that they missed the target.

Solution

To solve this problem, we can calculate the conditional probabilities for each group of shooters. The conditional probability is the probability of a shooter belonging to a specific group, given that they missed the target.

Let's calculate the conditional probabilities for each group:

- Group 1: 18 shooters with a probability of hitting the target of 0.8. - Group 2: 7 shooters with a probability of hitting the target of 0.7. - Group 3: 4 shooters with a probability of hitting the target of 0.6. - Group 4: 2 shooters with a probability of hitting the target of 0.5.

We can use Bayes' theorem to calculate the conditional probabilities. Bayes' theorem states that the conditional probability of an event A given event B can be calculated as:

P(A|B) = (P(B|A) * P(A)) / P(B)

In this case, event A represents the shooter belonging to a specific group, and event B represents the shooter missing the target.

Let's calculate the conditional probabilities for each group:

- Group 1: P(Group 1|Missed) = (P(Missed|Group 1) * P(Group 1)) / P(Missed) - Group 2: P(Group 2|Missed) = (P(Missed|Group 2) * P(Group 2)) / P(Missed) - Group 3: P(Group 3|Missed) = (P(Missed|Group 3) * P(Group 3)) / P(Missed) - Group 4: P(Group 4|Missed) = (P(Missed|Group 4) * P(Group 4)) / P(Missed)

To calculate the probabilities, we need to know the probability of missing the target, which is the complement of hitting the target. We can calculate it as:

P(Missed) = 1 - P(Hit)

Let's calculate the probabilities:

- P(Missed) = 1 - P(Hit) = 1 - 0.8 = 0.2

Now, let's calculate the conditional probabilities for each group:

- Group 1: P(Group 1|Missed) = (P(Missed|Group 1) * P(Group 1)) / P(Missed) = (0.2 * 18) / 0.2 = 18 - Group 2: P(Group 2|Missed) = (P(Missed|Group 2) * P(Group 2)) / P(Missed) = (0.3 * 7) / 0.2 = 2.45 - Group 3: P(Group 3|Missed) = (P(Missed|Group 3) * P(Group 3)) / P(Missed) = (0.4 * 4) / 0.2 = 2 - Group 4: P(Group 4|Missed) = (P(Missed|Group 4) * P(Group 4)) / P(Missed) = (0.5 * 2) / 0.2 = 5

Based on these calculations, the shooter is most likely to belong to Group 1, as it has the highest conditional probability of 18.