Bad apple

Probability Level 3

A basket contains 20 apples and 10 oranges out of which 5 apples and 3 oranges are defective.If a person takes out 2 at random what is the probability that either both are apples or both are good? Give your answer to 3 decimal places.

The answer is 0.726.

4 solutions

Pawan Kumar
Apr 8, 2015

$P(A) =$ Probability that both are APPLE $= \frac{20}{30} \times \frac{19}{29}$

$P(G) =$ Probability that both are GOOD $= \frac{22}{30} \times \frac{21}{29}$

$P(G \cap A) =$ Probability that both are GOOD APPLE $= \frac{15}{30} \times \frac{14}{29}$

$P(A \cup G) =$ Probability that both are APPLE or both are GOOD $= P(A) + P(G) - P(G \cap A)$

$= \frac{20}{30} \times \frac{19}{29} + \frac{22}{30} \times \frac{21}{29} - \frac{15}{30} \times \frac{14}{29} = 0.726$

Trevor Arashiro
Apr 8, 2015

Some one please verify the text below the soluion, the approximation section idk why it works and comes so close.

Call good apples $a$ and bad apples $a'$ .

Call good oranges $b$ and bad oranges $b'$

Denote $[xy]$ as the number of ways in which you can chose items $x$ and $y$ .

We need to find $\dfrac{[ab]+[aa]+[aa']+[a'a']+[bb]}{\dbinom{30}{2}}$

The sum of $[aa]+[aa']+[a'a']=\dbinom{20}{2}$

$[bb]=21$

$[ab]=105$

Thus we have

$\dfrac{190+21+105}{435}\approx .726$

Besides the typo at the end, (you should have .726 rather than .762), this looks good. I like the notational device with the primes and square brackets; makes the process seem more intuitive. :)

Brian Charlesworth - 6 years, 2 months ago

Boy am I good at making simple errors :3

I didn't type the approximation section because I posted this solution super late at night and I realized just how dumb it was in the morning. The value was super close but the method was totally wrong. It was pure coincidence lol.

Speaking of which. Do you believe in coincidence? Or are there just some unexplainable things the world may never know?

Trevor Arashiro - 6 years, 2 months ago

Well, I suppose that there are two distinct questions here. The Uncertainty Principle guarantees that there are some things "the world may never know". The quantum reality is that we don't even live in a universe which is governed strictly by causality; as Feynman once said, "sometimes things happen just because they can". So given this, can we determine whether "coincidence" is or is not possible? No. I would put it in the category of "unknowable" rather than "unexplainable". Improbable concurrences happen because they can, and when they do they make the news and everyone goes "Wow! How amazing! That can't be just a coincidence!", ignoring the fact that the more probable scenarios play out ad infinitum and receive no attention. The instinct our species has for probability is warped, and leaves us susceptible to irrational conclusions. Having said that, I do hold stock in Carl Jung's notions of the "collective unconscious", which holds that what seems coincidental at one level in fact has a deeper vein of connectivity.

Sigh .......

Short answer: yes, no, maybe so ..... :)

Brian Charlesworth - 6 years, 2 months ago

@Brian Charlesworth – "Yes, no, maybe so" sounds like every one of my answers on a history test :3.

I do believe that there is such a thing as coincidence but more than often it's not. No matter how unrelated two events may seem, they can be related. While one may not necessarily depend on the other to happen, the two may be directly or indirectly related. This is mainly due to the infinitely small amount of knowledge that humans poses. For example: full moons cause crime rates to go up; the two are completely unrelated. Well, what if the light reflected causes certain chemicals to be produced in the sectors of the brain most commonly associated with crime thought process.

Another great source of seemingly unexplainable coincidences is the placebo effect. I don't believe that good luck charms actually hold up to all the claims that their sellers make. However, just the presence of the good luck charm may be enough to provide a little edge of confidence. In fact, the placebo effect has been effective in curing many illnesses that to this day cannot be cured strictly through medical procedures.

While thinking about this, the biggest question my thought posed to me was, "Are there some events which in theory can not be related, have some sort of effect on eachother?" There is only one example that I could think of (the other is that future events affect past events, but we haven't disproven time travel yet thought it's highly unlikely). Say that right now (note that the word "now" kinda disregards time) on Planet X there is some event happening. That planet exists 10 light years away from ours. Can that event affect something happening now on our planet?

Trevor Arashiro - 6 years, 2 months ago

@Trevor Arashiro – Quantum entanglement, which Einstein disparaged as "spooky action at a distance", can (potentially) link the actions of two particles over great distances. If we were to somehow be observing these two particles simultaneously in their disparate locations and be unaware of their quantum "link", then would we conclude that their temporally similar behavior was just coincidence, or would we speculate on some deep, unseen connection? If the former, then we have missed an opportunity for furthering our knowledge; if the latter, then how would we go about proving our hypothesis?

You're right - there is so much we don't know that we would be remiss to discard the possibility of a correlation between seemingly unrelated events. But we need to temper our ambitions with the old dictum that "correlation does not imply causation". The perception of coincidence is dependent on the observer, and as with all things it is impossible to entirely remove ourselves, as observers, from the equation. If we choose to be subjective, we will likely believe in fate - a "Grand Plan", if you will - in which case there is no room for coincidence. But if we choose to be objective, then "everything is permitted", as Dostoevsky (so dramatically) said, in which case we have to keep our minds open to all possibilities, no matter how far-fetched they may at first seem to be. This includes the possibility of the future affecting the past, for we still don't really have a thorough understanding of what time actually is. (My favorite definition is that "time is what keeps everything from happening at once".)

O.k., I am now officially rambling. Your turn for more words of wisdom. :)

Brian Charlesworth - 6 years, 2 months ago

Abhishek Mb
Apr 6, 2015

Number of ways to pick two fruits A=30C2

Number of ways to pick two apples B= 20C2

Probability that both are apples P(x)=B/A

Number of ways to pick two fruits from 22 good fruits C=22C2

Probability that both are good P(y)=C/A

Number of good apples 20-5=15

Number of ways of picking 2 apples from good apples D=15C2

Probability that both are good and both are good P(z)=D/A

Final probability =P(x)+P(y)-P(z)

Ranik Chakraborty
Apr 5, 2015

Out of 30 items,two can be selected in 30C2 ways.So, total number of elementary events=30C2.Now,considering following cases(events): A=getting two apples;B=getting two good items. Required probability= P(AUB)=P(A)+P(B)-P(A intersection B).There are 20 apples, out of which 2 can be drawn in 20C2 ways. Hence, P(A)=20C2/30C2 . There are 8 defective pieces and the remaining 22 are good. Out of 22 good pieces, two can be selected in 22C2 ways.Hence,P(B)= 22C2/30C2 .Since there are 15 pieces which are good apples out of which 2 can be selected in 15C2 ways. Therefore,P(A intersection B)=15C2/30C2. Hence,required probability=(20C2/30C2) + (22C2/30C2) - (15C2/30C2)=316/435=0.726

Bad apple

The answer is 0.726.

4 solutions

0 pending reports