What is the probability that a fair die rolled once will land on either 4 or 5?
Andrew Henderson 
I'm not a native speaker so could someone please explain what either means in this situation please?
Does it mean (1) a fair die will land on 4 or on 5, (2) a fair die will not land on 4 or on 5 or (3) a fair die will not land on both 4 and 5?
By the way, please let me know if my math is correct for each situation:
(1) 1/6 + 1/6
(2) 1 - (1/6 + 1/6)
(3) 1 - 1/6 x 1/6
Thank you, I really appreciate you help!
$\endgroup$ 53 Answers
$\begingroup$As requested in the comments:
Your first reading is correct and your computation for its probability is also correct.
Your second reading would be consistent with the phrase "the fair die will land on $\textit {neither} 4 \textit {nor} 5" and, again, the calculation you give for that event is correct.
Your third reading doesn't really make sense or, if you read it literally, it is a certainty. A fair die can not land on two different numbers at once, so the probability that it fails to land on both $4$ and $5$ is $1$. To justify your computation, you'd have to have an event like "you roll the die twice, getting a $4$ the first time and a $5$ the second time."
$\endgroup$ $\begingroup$Put simply, it means (1) a fair die will land on 4 or on 5. Since this is 2 possibilities, out of a total of 6 on a die, your answer would be 2/6 = 1/3 (this is also equal to your 1/6 + 1/6.
As for (2), the question would state what you said "a fair die will not land on 4 or on 5" if this were the case. Your math is again, correct, but since 1-(1/6+1/6) = 4/6, you should always simplify to 2/3.
As pointed out in the comments, option (3) doesn't make sense. A die lands on 1 number, not 2 at the same time, so it can't land on both 4 and 5 and the same time.
Hopefully this helps.
$\endgroup$ $\begingroup$This is really a question of English rather than mathematics. Many non-native and even some native speakers get confused. In day to day life, the intended meaning is usually obvious but in mathematics, more care and precision is required. (Probably also in law but I am not a lawyer.)
Probability that the die will land on either $4$ or $5$ - $\frac{1}{3}$.
(The either is common but does not change the meaning much. You could omit it.)
Probability that the die will land on both $4$ and $5$ - $0$ (unless your die might balance on an edge).
(The both is common but does not change the meaning much. You could omit it.)
When negated it gets more confusing.
Probability that the die will not land on either $4$ or $5$ - $\frac{2}{3}$.
Probability that the die will not land on both $4$ and $5$ - $1$ (again, unless your die might balance on an edge).
An alternative way to say the first is: the die will land on neither $4$ nor $5$. I suggest avoiding that if you are not familiar with it.
For non-mathematical uses.
In the Philippines, I saw a sign which said: "No parking on both sides of the road". I considered parking and arguing that I had only parked on one side of the road.
If someone says: "would you like tea or coffee?", they probably do not expect the answer "both" but in mathematics "or" is inclusive unless stated otherwise. You could say: "would you like either tea or coffee?" but that is not common in a question like that.
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