Dice Probability: Product as Perfect Square
When two dice are thrown simultaneously, each die has 6 possible outcomes (numbers 1 through 6). The total number of possible outcomes is the product of the outcomes for each die.
Total possible outcomes = $6 \times 6 = 36$.
Identifying Favorable Outcomes
We need to find the outcomes where the product of the numbers on the two dice ($d_1 \times d_2$) is a perfect square. A perfect square is a number that is the square of an integer (e.g., 1, 4, 9, 16, 25, 36, ...).
Let's list the pairs $(d_1, d_2)$ such that $d_1 \times d_2$ is a perfect square, where $d_1, d_2 \in \{1, 2, 3, 4, 5, 6\}$:
- If the product is 1: (1, 1)
- If the product is 4: (1, 4), (4, 1), (2, 2)
- If the product is 9: (3, 3)
- If the product is 16: (4, 4)
- If the product is 25: (5, 5)
- If the product is 36: (6, 6)
The set of favorable outcomes is {(1, 1), (1, 4), (4, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}. The total number of favorable outcomes is 8.
Calculating the Probability
Probability is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.
$ P(\text{Product is a perfect square}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} $
$ P(\text{Product is a perfect square}) = \frac{8}{36} $
Simplifying the fraction gives:
$ \frac{8}{36} = \frac{2 \times 4}{9 \times 4} = \frac{2}{9} $
The probability that the product of the numbers appearing on the top faces of the dice is a perfect square is 2/9.
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. 1/9Statements 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.
In 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:
