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CAT Exam  >  CAT Questions  >   Three boxes A, B, C each contain 4 white bal... Start Learning for Free
Three boxes A, B, C each contain 4 white balls and 5 black balls. All the balls are identical except for the colour. A ball is shifted from box A to box B and then a ball is shifted from box B to box C and finally a ball is shifted from box C to box A. The probability that each of the boxes will contain again 4 white balls and 5 black balls is
  • a)
    14/45
  • b)
    4/5
  • c)
    16/25
  • d)
    1/45
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
Three boxes A, B, C each contain 4 white balls and 5 black balls. All...
Even after shifting the ball, if the boxes contain same number of black and white balls, then it can happen in two ways.
Either black ball is shifted from box A to box B and then from box B to box C and then from box C to box A, so that the same number of black and white balls remain in the boxes.
Or white ball is shifted from box A to box B, then from box B to box C and then box C to box A
Hence, we will consider each case to find the probability
Case 1: Black ball is shifted from box A to box B(5/9) and then from box B to box C(6/10) and then from box C to box A(6/10)
Case 2: white ball is shifted from box A to box B ( 4 /9), then from box B to box C(5/10) and then box C to box A(5/10)
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Community Answer
Three boxes A, B, C each contain 4 white balls and 5 black balls. All...
Problem Analysis:
We have three boxes A, B, and C, each containing 4 white balls and 5 black balls. We are shifting one ball from box A to box B, then one ball from box B to box C, and finally one ball from box C to box A. We need to find the probability that each box will contain 4 white balls and 5 black balls after these shifts.

Solution:
Let's analyze the possible scenarios after each shift:

After the first shift:
- Box A: 3 white balls and 5 black balls
- Box B: 5 white balls and 5 black balls

After the second shift:
- Box B: 4 white balls and 4 black balls
- Box C: 5 white balls and 5 black balls

After the third shift:
- Box A: 4 white balls and 4 black balls
- Box C: 4 white balls and 6 black balls

In order for each box to have 4 white balls and 5 black balls after the shifts, we need to consider the following possibilities:

1. The ball shifted from box C to box A is a white ball:
- Probability of this event = (4 white balls in box C / 10 total balls in box C) = 2/5
- After this shift, we have:
- Box A: 5 white balls and 4 black balls
- Box C: 3 white balls and 6 black balls

2. The ball shifted from box C to box A is a black ball:
- Probability of this event = (6 black balls in box C / 10 total balls in box C) = 3/5
- After this shift, we have:
- Box A: 4 white balls and 5 black balls
- Box C: 4 white balls and 5 black balls

The probability that each box will contain 4 white balls and 5 black balls is the sum of the probabilities of the above two possibilities:

Probability = (2/5) * (3/4) + (3/5) * (1/4) = 6/20 + 3/20 = 9/20

Therefore, the correct answer is option 'A' (14/45).

Answer:
The probability that each of the boxes will contain again 4 white balls and 5 black balls is 14/45.
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Directions: The treatment of probability has been confused by mathematicians. From the beginning there was an ambiguity in dealing with the calculus of probability. When the Cehvalier de Mere consulted Pascal on the problems involved in the games of dice, the great mathematician should have frankly told his friend the truth, namely, that mathematics could not be of any use to the gambler in a game of pure chance. Instead he wrapped his answer in the symbolic language of mathematics. What could easily be explained in a few sentences of mundane speech was expressed in a terminology which was unfamiliar to the immense majority and therefore regarded with reverential awe. People suspected that the puzzling formulae contain some important revelations, hidden to the uninitiated; they got the impression that a scientific method of gambling exists and that the esoteric teachings of mathematics provide a key to winning. The heavenly mystic Pascal unintentionally became the patron saint of gambling. The textbooks of the calculus of probability gratuitously propagandize for the gambling casinos precisely because they are sealed books to the layman.There are two entirely different instances of probability; we may call them class probability (or frequency probability) and case probability (or the specific understanding of the sciences of human action). The field for the application of the former is the field of the natural sciences, entirely ruled by causality; the field for the application of the latter is the field of the sciences of human action, entirely ruled by teleology.Class probability means: We know or assume to know, with regard to the problem concerned, everything about the behavior of a whole class of events or phenomena; but about the actual singular events or phenomena we know nothing but that they are elements of this class. We know, for instance, that there are ninety tickets in a lottery and that five of them will be drawn. Thus we know all about the behavior of the whole class of tickets. But with regard to the singular tickets we do not know anything but that they arc elements of this class of ticketsCase probability means: We know, with regard to a particular event, some of the factors which determine its outcome; but there are other determining factors about which we know nothing. Case probability has nothing in common with class probability but the incompleteness of our knowledge. In every other regard the two are entirely differentThere are, of course, many instances in which men try to forecast a particular future event on the basis of their knowledge about the behavior of the class. A doctor may determine the chances for the full recovery of his patient if he knows that 70 per cent of those afflicted with the same disease recover. If he expresses his judgment correctly, he will not say more than that the probability of recovery is 0.7, that is, out of ten patients not more than three on the average die. All such predictions about external events, i.e., events in the field of the natural sciences, are of this character. They are in fact not forecasts about the issue of the case in question, but statements about the frequency of the various possible outcomes. They arc based either on statistical information or simply on the rough estimate of the frequency derived from nonstatistical experience.As far as such types of probable statements are concerned, we are not faced with case probability. In fact we do not know anything about the case in question except that it is an instance of a class the behavior of which we know or think we know.A surgeon tells a patient who considers submitting himself to an operation that thirty out of every hundred undergoing such an operation die. If the patient asks whether this number of deaths is already full, he has misunderstood the sense of the doctor's statement. He has fallen prey to the error known as the "gambler's fallacy." Like the roulette player who concludes from a run often red in succession that the probability of the next turn being black is now greater than it was before the run, he confuses case probability with class probability.Which of the following best describes the key point made by the author?

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Three boxes A, B, C each contain 4 white balls and 5 black balls. All the balls are identical except for the colour. A ball is shifted from box A to box B and then a ball is shifted from box B to box C and finally a ball is shifted from box C to box A. The probability that each of the boxes will contain again 4 white balls and 5 black balls isa)14/45b)4/5c)16/25d)1/45Correct answer is option 'A'. Can you explain this answer?
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