Statements followed by some conclusions are given below. Statements: 1. A bag has 2 white, 3 black, 4 red and 6 green balls. 2. 1 ball selected at random from the bag. Conclusions: I. The probability that a black ball is selected is 1/5 II. The probability that a red ball is selected is 6/15 Find which of the conclusions logically follows from the given statement A. Only conclusion I follows. B. Only conclusion II follows. C. Both I and II follow. D. Neither I nor II follows.
A
Understanding the Probability Problem
The question provides information about the number of balls of different colors in a bag and asks us to evaluate two conclusions regarding the probability of selecting a ball of a specific color at random.
The given statements are:
- A bag has 2 white, 3 black, 4 red and 6 green balls.
- 1 ball selected at random from the bag.
The conclusions to be checked are:
- Conclusion I: The probability that a black ball is selected is 1/5.
- Conclusion II: The probability that a red ball is selected is 6/15.
Calculating Probabilities from Statements
To find the probability of selecting a specific color ball, we first need to determine the total number of balls in the bag. The total number of outcomes when selecting one ball is equal to the total number of balls.
Total number of balls = Number of white balls + Number of black balls + Number of red balls + Number of green balls
Total number of balls = $2 + 3 + 4 + 6 = 15$ balls.
The formula for the probability of an event is:
Probability (Event) = $\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
Evaluating Conclusion I: Probability of Selecting a Black Ball
Conclusion I states that the probability of selecting a black ball is 1/5.
Number of black balls = 3
Total number of balls = 15
Probability of selecting a black ball = $\frac{\text{Number of black balls}}{\text{Total number of balls}} = \frac{3}{15}$
Simplifying the fraction $\frac{3}{15}$, we divide both the numerator and the denominator by their greatest common divisor, which is 3.
$\frac{3 \div 3}{15 \div 3} = \frac{1}{5}$
The calculated probability of selecting a black ball is 1/5. This matches Conclusion I.
Therefore, Conclusion I logically follows from the given statements.
Evaluating Conclusion II: Probability of Selecting a Red Ball
Conclusion II states that the probability of selecting a red ball is 6/15.
Number of red balls = 4
Total number of balls = 15
Probability of selecting a red ball = $\frac{\text{Number of red balls}}{\text{Total number of balls}} = \frac{4}{15}$
The calculated probability of selecting a red ball is 4/15. This does not match Conclusion II, which states the probability is 6/15.
Therefore, Conclusion II does not logically follow from the given statements.
Summary of Conclusions
Based on our calculations:
- Conclusion I: The probability that a black ball is selected is 1/5. (Calculated: 1/5) - Follows
- Conclusion II: The probability that a red ball is selected is 6/15. (Calculated: 4/15) - Does not follow
Only Conclusion I logically follows from the given statements.
| Ball Color | Number of Balls | Calculated Probability | Simplified Probability | Conclusion Statement | Conclusion Matches? |
|---|---|---|---|---|---|
| Black | 3 | $\frac{3}{15}$ | $\frac{1}{5}$ | 1/5 | Yes |
| Red | 4 | $\frac{4}{15}$ | $\frac{4}{15}$ | 6/15 | No |
Conclusion Derivation
Analysing the statements and conclusions, we find that only the first conclusion aligns with the probability calculated from the information provided in the statements.
Revision Table: Probability Concepts
| Concept | Description | Formula |
|---|---|---|
| Probability | A measure of the likelihood of an event occurring. | $\frac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}}$ |
| Random Selection | Each item has an equal chance of being selected. | Implies using the basic probability formula. |
| Simplifying Fractions | Reducing a fraction to its simplest form by dividing the numerator and denominator by their GCD. | $\frac{a}{b} = \frac{a \div \text{GCD}(a, b)}{b \div \text{GCD}(a, b)}$ |
Additional Information: Types of Probability
Probability can be interpreted in different ways, but for problems like this, we use classical probability.
- Classical Probability: This is used when all outcomes in the sample space are equally likely. It relies on counting the number of favorable outcomes and the total number of possible outcomes. This is the type of probability used in the ball selection problem.
- Empirical Probability: This is based on observed data from experiments or historical records. It is the ratio of the number of times an event occurred to the total number of trials.
- Subjective Probability: This is based on personal judgment, experience, or intuition. It is a personal estimate of the likelihood of an event.
In this specific question, the selection of a ball at random from the bag implies that each ball is equally likely to be chosen, making it a classical probability problem.
Important Questions from Probability
Two dice are thrown simultaneously and the sum of the numbers appearing on them is noted. What is the probability that the sum is 12?
If a box contains 3 white cushions, 4 red cushions and 5 blue cushions, what is the probability of selecting a white or blue cushion?
A. 2/3
B. 3/4
C. 1/4
D. 1/9In a shooting test, the probabilities of hitting the target are 1/2 for A, 2/3 for B and 3/4 for C. If they fire at the same target, what is the probability that only one of them hits the target?
A bag contains balls numbered from 1 to 42. One ball is drawn at random from these balls. The probability that its number is a multiple of 7 or 8 is:
A bag contains $3$ red balls, $4$ white balls, and $5$ green balls. Of these, three balls are drawn at random. What is the probability that exactly two of the balls drawn are of the same colour?
