Bunuel wrote:

Raffle tickets numbered consecutively from 101 through 350 are placed in a box. What is the probability that a ticket selected at random will have a number with a hundreds digit of 2 ?

(A) 2/5

(B) 2/7

(C) 33/83

(D) 99/250

(E) 100/249

Richlove wrote:

Please explain to me how the favorable cases inclusive are from 200 - 299 . Most especially the hundred digit of 2.

Richlove , I am not quite sure what you are asking.

Please be more specific next time?

I think your question involves inclusive counting.*

Favorable casesAny integer of the form 2 _ _ is a "favorable case"

How many of those 2 _ _ integers / terms are there?

• The first term in form "2 _ _" is 200

• The last term in the form "2 _ _" is 299

• Number of favorable cases? All the numbers from 200 to 299

• To find the number of favorable cases, use

inclusive counting formula:

(Last term - first term) PLUS ONESubtraction only, (Last - First), yields the

difference between integers

Subtraction does not yield the number of integers.

Take a small sample to see why

Small sampleHow many integers are there from 2 to 5?

2, 3, 4, 5: FOUR integers

But if we subtract? 5 - 2 = 3. Not correct

Add 1 to (5 - 2) = 3. Then (3 + 1) = FOUR integers

Counting: probabilityFor the problem, we need \(\frac{FavorableCases}{PossibleOutcomes}\)

Total possible outcomes?

All integers from 101 to 350

(350 - 101) =

249 + 1 =

250 all possible outcomes

(The box contains ALL the numbers: the 100s group, the 200s group, and the 300s group)

Favorable cases?

All the integers in the group with form 2 _ _

First term is 200, last term is 299

All integers in the 200s group?

(299 - 200) =

99 + 1 = 100 integers from 200 to 299

Probability?

Probability that you will pick a number from the 200s group?

\(\frac{Favorable}{Possible} = \frac{200s}{AllTickets'Numbers} = \frac{100}{250} = \frac{2}{5}\)

Answer C. Hope that helps.

*Other than "numbers that have hundreds digit of 2," what could be the "favorable cases" here? If you are asking why integers with 2 in the hundreds place are "favorable"? Because the prompt defines "numbers with a hundreds digit of 2" as "success."
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