# What does "Four selected students are not born in the same months" mean?

I am teaching on probability. I found a question that seems to be ambiguous as follows.

Four students are randomly chosen from a place. Assuming the birthdays of people are equally likely to occur in any month, find the probability that 4 selected students are not born in the same months.

The answer key of this question is $$1-\frac{12}{12^4}$$.

If I agree with this key using negation principle then I think the question must be rewritten as

Find the probability that at least one student of the selected students is not born in the same month.

Because the negation of "all students are born in the same month" is "at least one of the selected students is not born in the same month".

Thus for me, the original question "find the probability that 4 selected students are not born in the same months" is badly written.

# Question

What do you think? Is it also ambiguous for you? Is there any better expression for the original question to get rid of its ambiguity?

# Edit

Other examples:

• A = 4 balls are red.

A' = at least one ball is not red (rather than 4 balls are not red)

• A = 4 cars are sold.

A' = at least one car is not sold (rather than 4 cars are not sold)

• An example of two students who were not born in the same month: Alice was born on January 1, Bob was born on February 3. Can you give me an example of one student who was not born in the same month? What is that student's birthday? Oct 22, 2018 at 8:04
• @JoelReyesNoche: We are just talking about the month of DOB (data of birth). At least one student among the selected is not born in the same month: (Jan, Feb, Feb) , (Feb, Jan, Feb), (Mar, Mar, Dec), etc... too many to be listed here. Oct 22, 2018 at 8:07
• So you mean "at least one of the selected students is not born in the same month as the other selected students"? Oct 22, 2018 at 8:09
• If you want only cases such as [Jan, Feb, Mar, Apr] and not cases such as [Jan, Feb, Feb, Feb], then the expression could be "the four selected students are born on different months" or "none of the four selected students have the same birth month as the others." Oct 22, 2018 at 9:05
• @shoover: Corrected. Thanks. Oct 22, 2018 at 17:36

The original question is:

Four people are randomly chosen from a place. Assuming the birthdays of people are equally likely to occur in any month, find the probability that 4 selected students are not born in the same months.

I understand that the intended event includes cases such as [Jan, Feb, Mar, Apr] and [Jan, Feb, Feb, Feb] and excludes cases such as [Feb, Feb, Feb, Feb].

Your proposed revision is to edit the last clause to:

find the probability that at least one student of the selected students is not born in the same month.

In my opinion, both the original question and your proposed revision could be slightly edited to improve clarity:

Four students are randomly selected. Assuming that the students are equally likely to be born in any month, find the probability that the four selected students are not all born on the same month.

and

Four students are randomly selected. Assuming that the students are equally likely to be born in any month, find the probability that at least one of the four selected students is not born on the same month as another selected student.

• For the last, "as the other selected students" is not necessarily needed. Oct 22, 2018 at 9:07
• "same month as the other students" somehow implies/suggests that they do share a common month. Oct 22, 2018 at 14:07
• @Jasper, thanks, I've edited my answer. Oct 22, 2018 at 22:20