A class with n kids lines up for recess. The order in which the kids line up is random with each ordering being equally likely. There are three kids in the class named Hubert, Celia and Felicity.
The use of the word "or" in the description of the events, should be interpreted as the inclusive or. That is "A or B" means that A is true, B is true or both A and B are true.

Give an expression for each of the probabilities below as a function of n. Simplify your final expression as much as possible so that your answer does not include any expressions of the form (ab)(ab).

(a)What is the probability that Celia is first in line or Felicity is first in line?

(b)What is the probability that Celia is first in line or Felicity is last in line?

(c)What is the probability that Celia is not next to Felicity in line?

(d)What is the probability that Hubert is next to Celia or Felicity in line?

c. P(([tex]F^{c}[/tex] ∩ C/[tex]F^{c}[/tex]) ∪ (C ∩ [tex]F^{c}[/tex]/C)) = P([tex]F^{c}[/tex] ∩ C/[tex]F^{c}[/tex]) + P(C ∩ [tex]F^{c}[/tex]/C) = 2 * P([tex]F^{c}[/tex] ∩ C/[tex]F^{c}[/tex])

d. P(H ∩ (C ∪ F)) = P(H) * ((P(C) + P(F) - P(C ∩ F))

Step-by-step explanation:

Hello!

You have a sample of n kids lining up for recess.

The order in which they line up is random with each ordering bein equally likely, this means that the probability of each kid to take a position is [tex] \ frac {1} {n (total of kids/positions)} [/ tex ]

You are being asked about 3 kids from the class, I'll assign a letter to each kid:

H: Hubert

C: Celia

F: Felicity

to. What is the probability that Celia is first in line or Felicity is first in line?

Translated to symbols this probability is:

P (C₁ ∪ F₁)

Where the subfix 1 indicates that the girls are first in the line and the mathematical symbol "∪" indicates the theoretical union between the two events, colloquially used "or". (in other words, when the sentence mentions one "or" other event, you symbolize it as the union "∪" of both events.)

P (C₁ ∪ F₁) = P (C₁) + P (F₁) - P (C₁ ∩ F₁)

Where P (C₁ ∩ F₁) is the intersection between both events. This union takes importance when the events aren't mutually exclusive (that can happen at the same time). Since either Celia or Felicity can be first on the line but not both at the same time there is no intersection between the two events, so P (C₁ ∩ F₁) = 0

And the probability is:

P (C₁ ∪ F₁) = P (C₁) + P (F₁)

b. What is the probability that Celia is first in line or Felicity is last in line?

In this case, the events are not mutually exclusive, since it can happen that "Celia is first but Felicity is not last" or "Felicity is last but Celia is not first" or "Celia ist first in line and Felicity is last", so there is an intersection between them.

The probability is symbolized as

P (C₁ ∪ Fₙ) = P (C₁) + P (Fₙ) - P (C₁ ∩ Fₙ) = P (C₁) + P (Fₙ) - (P (C₁) * P (Fₙ / C₁))

Where the suffix 1 refers to the first position and n refers to the last position.

And P (C₁ ∩ Fₙ) = P (C₁) * P (Fₙ / C₁) because the events "Celia" and "Felicity" are not independent. (Every time a kid takes his place the probability of the next one is affected)

c. What is the probability that Celia is not next to Felicity in line?

In this case, you have a new event that is the complement of Felicity being there, F. I'll symbolize the event of "not Felicity" or "any other kid but Felicity" as [tex] F ^ {c} [/ tex]

In this point two options could happen, one, that Celia takes her place and the kid before her is not Felicity, or, two, that Celia takes her place and the kid after her is not Felicity.

P (([tex] F ^ {/ tex] ∩ C / [tex] F ^ {c} / / tex]) ∪ (C ∩ [tex] F ^ {c} / / tex] / C) ) = P ([tex] F ^ {/ tex] ∩ C / [tex] F ^ {c / [tex]) + P (C ∩ [tex] F ^ c [/ tex] / C) = 2 * P ([tex] F ^ [/ tex] ∩ C / [tex] F ^ [/ tex])

d. What is the probability that Hubert is next to Celia or Felicity in line?

In this case, it can happen that "Hubert is next to Celia but not Felicity" or "Hubert is next to Felicity but not Celia" or "Hubert is next to Celia and Felicity"

P (H ∩ (C ∪ F)) = P (H) * ((P (C) + P (F) - P (C ∩ F))

I hope it helps!