Truths and Lies

Probability Level 2

Alex tells the truth 70% of the time while Blake tells the truth 80% of the time, independently of each other. They are shown an apple and asked "Is it an apple or a banana?". What is the probability that they will give different answers?

0.4 0.56 0.38 0.42

10 solutions

Kunal Jain
May 19, 2015

To contradict each other, we have one of two possibilities:

Alex tells the truth and Blake lies: $70\% \cdot 20\% = 14\%$
Alex lies and Blake tells the truth: $30\% \cdot 80\% = 24\%$

Adding up these two cases, we have a total probability of 38%.

I don't understand why the percentages are multiplied and the solutions are added? I get how you did the math, I just don't understand the method. Explain?

The way I see it is that there are 70 times they give the same correct answer, 20 times they give the same wrong answer, leaving 10 times they give different answers. So, my answer would have be 10%.

Jeff Carter - 4 years, 8 months ago

Check the ‘Rule of Product’ wiki

Kano Boom - 1 year, 8 months ago

thank you , you totally confused me

Rushikesh Kohre - 3 years, 10 months ago

Alex Delhumeau
May 18, 2015

The probability that both A and B speak true is $.7\times{.8}$ while the probability that they both lie is $.3\times{.2}$ . Therefore, taking the complement will give the probability that they have contradicted each other: $1-0.56-0.06=\boxed{0.38}$ .

parentheses around the first two terms would have clarified your solution.

Sya Lark - 4 years, 3 months ago

Brandon Kinsey
May 18, 2015

If they both lie, isn't it possible they would contradict each other?

My understanding is that the question is equating one of them lying and the other one telling the truth as "contradiction."

Alex Delhumeau - 6 years ago

Not if we assume the answer is binary

Jordan Webber - 6 years ago

Its the same event so no that would not be possible.

Adhiraj Chattopadhyay - 6 years ago

If both lie, they'll describe the same event in the same way. Therefore, they won't contradict each other.

Shawn Pereira - 6 years ago

Need not be. If A says that sky is red and B says that sky is green, both are lying but they still contradict each other.

Vijay Bhaskar - 6 years ago

I agree. But in this case, we're made to assume that there's only one lie and one true way of describing something.

Shawn Pereira - 6 years ago

Good one :-)

Vijay Bhaskar - 6 years ago

In response to Brandon Kinsey, I don't think so, because I think the question is assuming that person A and person B speak the truth or lie about the same event. For example, if somehow both A and B know that today will be a rainy day and if they both lie, then both will say, 'it won't rain today,' and so they are not contradicting each other. Edit: It also depends on what is being asked. Asking 'What will be the weather today?' and asking, 'Will it rain today?' could elicit different responses.

Aviral Jain - 6 years ago

However A may lie saying it will not rain today and B may lie saying it will snow today, these are still contradicting eachother even though both are lying

Kayla Kennedy - 6 years ago

Yes, you are right. Then, I guess another assumption that would clarify this is that there is only one lie that can be told. And one more thing, what you have said depends on what was asked. If someone asks, 'Will it rain today?' then saying , 'it will snow today' does not make any sense. Whereas, saying, 'Yes, it will rain today' or 'No, it will not rain today' is a more sensible answer.

Aviral Jain - 6 years ago

I agree with u your exp. is easily understandable

Huawei Star - 5 years ago

The choice given them is whether it's an apple or a banana, so they only have a binary option. If they lie, they must say "banana". If they tell the truth, they must say "apple".

Brian Egedy - 5 years, 2 months ago

Easy to slip up when bananas are around😂

Steve Lewis - 4 years ago

Gabriel Páez
Jul 2, 2015

If they contradict each other then one is telling truth and the other is lying (Logic pls), then you must sum the possibilities of each one lying given that the other is telling truth, so, if A is telling truth ( 70% $=0.7$ ) and B has 20% ( 20% = 0.2) prob of lying then the probability of A contradicting B is $0.7 * 0.2 = 0.14$ , and if B is telling truth ( 80% $=0.8$ ) and A has 30% ( 30% = 0.3 ) prob of lying then the probability of B contradicting A is $0.8 * 0.3 = 0.24$ , then you sum both's probability and get $0.14 + 0.24 = 0.38$

Lucy Nech
May 24, 2015

Perhaps explicitly stating that A's actions are independent of B's actions would improve the problem statement?

Ameya Patil
May 24, 2015

when A speaks the truth B lies and vice versa.... that is contradicting..

So,

7/10 * 2/10 + 8/10 * 3/10 = 0.38.........

Steve Zagieboylo
Jun 4, 2017

I disagree. They might both decide to tell the truth, but Alex says, "It's an apple" while Blake, who has studied boolean algebra, simply says, "yes." These are different answers.

Saya Suka
Feb 16, 2021

Answer
= 1 - (0.7) x (0.8) - (1 - 0.7) x (1 - 0.8)
= 1 - 0.56 - 0.06
= 1 - 0.62
= 0.38

Darlyn Daniel Victoriano Bremont
Jan 27, 2019

Un Owen
May 21, 2018

They are shown an apple. The probability that Alex tells the truth, and that Blake lies is .7 x .2=.14. The probability that Alex lies, and that Blake tells the truth is .3 x .8, or .24. The probability that their answers differ is .14 + .24= .38

Truths and Lies

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