# Why is P(X>15) = P(Y ≤ 4)?

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#1
I have a question from the S2 Edexcel book and the answer:

In a cafe, 70% of customers buy a cup of tea. In a random sample of 20 customers find the probability that more than 15 buy a cup of tea.

X~B(20, 0.7)
Since 0.7 is not in the tables you will need to consider the complementary random variable Y.
Y~B(20, 0.3)
P(X>15) = P(Y ≤ 4) = 0.2375

I don't understand why X>15 is the same as Y ≤ 4.
0
5 years ago
#2
(Original post by кяя)
I have a question from the S2 Edexcel book and the answer:

In a cafe, 70% of customers buy a cup of tea. In a random sample of 20 customers find the probability that more than 15 buy a cup of tea.

X~B(20, 0.7)
Since 0.7 is not in the tables you will need to consider the complementary random variable Y.
Y~B(20, 0.3)
P(X>15) = P(Y ≤ 4) = 0.2375

I don't understand why X>15 is the same as Y ≤ 4.
Assuming X is the number of people who buy tea and Y is the number of people who do not buy tea, then If more then 15 people buy tea, then less than or equal to 4 did not. To understand it, consider the possible outcomes if X > 15:

If X = 16 then Y = 20 - 16 = 4
X = 17 then Y = 20 - 17 = 3
X = 18 then Y = 2
X = 19 then Y = 1
X = 20 then Y = 0

So clearly Y <= 4 if X > 15
1
5 years ago
#3
Because you've changed to 0.3 (I find it easier to change the Y variable to X for example so you don't get confused), it means you are now testing the probability for the those who didn't buy a cup of tea opposing to who did.

Therefore, because you want to find out greater than 15, when you change from 0.7 to 0.3, (X< or = 4) is the same as (Y>15) as you are testing the probability that 4 or less people didn't buy a cup of tea (By extension meaning that more than 15 people did buy a cup of tea). The 16, 17, 18, 19 and 20th person(s) are who you are testing (if they bought a cup of tea or not)

I hope that makes sense, ask me more questions if it doesn't

Posted from TSR Mobile
1
#4
(Original post by 16Characters....)
Assuming X is the number of people who buy tea and Y is the number of people who do not buy tea, then If more then 15 people buy tea, then less than or equal to 4 did not. To understand it, consider the possible outcomes if X > 15:

If X = 16 then Y = 20 - 16 = 4
X = 17 then Y = 20 - 17 = 3
X = 18 then Y = 2
X = 19 then Y = 1
X = 20 then Y = 0

So clearly Y <= 4 if X > 15
Thank you so much for your answer; I can't believe I completely forgot how to do this. 0
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