Take two long dice (rectangular parallelepiped), each having four rectangular faces labelled as 2, 3, 5, and 7. If thrown, the long dice cannot land on the square faces and has 1/4 probability of landing on any of the four rectangular faces. The label on the top face of the dice is the score of the throw.
If thrown together, what is the probability of getting the sum of the two long dice scores greater than 11?
Understanding the Dice Probabilities
We are given two identical long dice. Each die has four rectangular faces labeled with scores: 2, 3, 5, and 7.
The dice cannot land on the square faces. This means each of the four labeled faces has an equal probability of being the top face when a die is thrown.
The probability of landing on any specific face (2, 3, 5, or 7) for a single die is $P(\text{face}) = \frac{1}{4}$.
Calculating Total Possible Outcomes
When two such dice are thrown together, the total number of possible outcomes is the product of the number of outcomes for each die.
Total Outcomes = (Number of faces on Die 1) $\times$ (Number of faces on Die 2)
Total Outcomes = $4 \times 4 = 16$.
Each of these 16 outcomes is equally likely because the individual face probabilities are equal.
Identifying Favorable Outcomes (Sum > 11)
We need to find the pairs of scores (Die 1, Die 2) whose sum is greater than 11.
Let's list the possible pairs systematically:
- If Die 1 = 2, the required score on Die 2 must be > 9 (impossible with faces {2, 3, 5, 7}).
- If Die 1 = 3, the required score on Die 2 must be > 8 (impossible).
- If Die 1 = 5, the required score on Die 2 must be > 6. The only possibility is Die 2 = 7. Pair: (5, 7).
- If Die 1 = 7, the required score on Die 2 must be > 4. Possible scores are Die 2 = 5 or Die 2 = 7. Pairs: (7, 5), (7, 7).
The pairs of scores resulting in a sum greater than 11 are (5, 7), (7, 5), and (7, 7).
Number of Favorable Outcomes = 3.
Determining the Probability
The probability of an event is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.
Probability (Sum > 11) = $\frac{\text{Number of Favorable Outcomes}}{\text{Total Possible Outcomes}}$
Probability (Sum > 11) = $\frac{3}{16}$.
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:
