8. A bag contains 8 square and 3 spherical objects, and two objects are drawn one
after another. If the first drawn object is put back in the bag before the second object
is drawn, the probability that both are of the same shape is x. If the first drawn object
is not put back in the bag, then the probability that both are of the same shape is
a. smaller than x
b. equal to x
c. greater than x
d. greater than or smaller than x, depending on which shape is drawn first
Probability Bag 8 Squares 3 Spheres
With replacement, the probability both objects are the same shape (x) equals 0.603. Without replacement, this probability drops to 0.564, making it smaller than x. The correct answer is option a. smaller than x.
In probability problems involving a bag with 8 squares and 3 spheres, understanding draws with replacement versus without replacement reveals key differences in same-shape outcomes. This CSIR NET-level question tests conditional probability concepts crucial for exams. The probability both drawn objects match in shape (both squares or both spheres) shifts based on replacement, yielding precise comparisons.
Problem Setup
A bag holds 11 objects: 8 squares and 3 spheres. Draws occur sequentially, and “same shape” means both squares or both spheres.
With Replacement (x)
Probabilities remain independent as the first object returns.
- Both squares: 8/11 × 8/11 = 64/121
- Both spheres: 3/11 × 3/11 = 9/121
- Total: x = 73/121 ≈ 0.603
Without Replacement
Probabilities depend on the first draw, reducing total objects to 10.
- Both squares: 8/11 × 7/10 = 56/110
- Both spheres: 3/11 × 2/10 = 6/110
- Total: 62/110 = 31/55 ≈ 0.564
Calculations Breakdown
| Scenario | Both Squares | Both Spheres | Total Probability |
|---|---|---|---|
| With Replacement | 64/121 | 9/121 | 73/121 ≈ 0.603 |
| Without Replacement | 56/110 | 6/110 | 62/110 ≈ 0.564 |
Option Analysis
a. smaller than x (Correct)
Without replacement lowers same-shape odds (0.564 < 0.603) due to fewer matching objects left after the first draw.
b. equal to x
Incorrect; independence fails without replacement, creating dependency between draws.
c. greater than x
Incorrect; dependency reduces probability of matching shapes in this scenario.
d. greater than or smaller than x, depending on which shape is drawn first
Incorrect; overall probability stays smaller regardless of first draw outcome.
Why Smaller Without Replacement?
- Fewer objects post-first draw reduce matching chances, especially for scarce spheres (3/11 → 2/10).
- This holds universally here, eliminating conditional variations.
- For CSIR NET aspirants, master such bag-and-ball scenarios via tree diagrams and fractions.
