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
A
Calculating Probability: White or Blue Cushion Selection
This problem asks us to find the probability of selecting either a white cushion or a blue cushion from a box containing different colored cushions. To solve this, we need to understand the total number of outcomes and the number of favorable outcomes for each event.
Understanding the Cushion Contents
The box contains:
- White cushions: 3
- Red cushions: 4
- Blue cushions: 5
Calculating Total Possible Outcomes
The total number of cushions in the box is the sum of the cushions of all colors.
Total cushions = Number of white cushions + Number of red cushions + Number of blue cushions
Total cushions = $3 + 4 + 5 = 12$
So, there are 12 total possible outcomes when selecting one cushion from the box.
Identifying Favorable Outcomes
We are interested in the probability of selecting a white or a blue cushion. This means our favorable outcomes are the number of white cushions plus the number of blue cushions.
Number of favorable outcomes (white or blue) = Number of white cushions + Number of blue cushions
Number of favorable outcomes (white or blue) = $3 + 5 = 8$
Calculating Individual Probabilities
Probability is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.
The probability of selecting a white cushion, denoted as P(White), is:
P(White) = $\frac{\text{Number of white cushions}}{\text{Total cushions}} = \frac{3}{12}$
The probability of selecting a blue cushion, denoted as P(Blue), is:
P(Blue) = $\frac{\text{Number of blue cushions}}{\text{Total cushions}} = \frac{5}{12}$
Applying the 'OR' Rule for Probability
When we want the probability of one event or another event happening, and these events cannot happen at the same time (they are mutually exclusive), we add their individual probabilities. Selecting a white cushion and selecting a blue cushion in a single draw are mutually exclusive events.
The probability of selecting a white or a blue cushion, denoted as P(White or Blue), is:
P(White or Blue) = P(White) + P(Blue)
P(White or Blue) = $\frac{3}{12} + \frac{5}{12}$
Adding the Probabilities
To add fractions with the same denominator, we add the numerators and keep the denominator the same.
P(White or Blue) = $\frac{3 + 5}{12} = \frac{8}{12}$
Simplifying the Result
The fraction $\frac{8}{12}$ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4.
$\frac{8 \div 4}{12 \div 4} = \frac{2}{3}$
Therefore, the probability of selecting a white or blue cushion is $\frac{2}{3}$.
Conclusion
Based on our calculation, the probability of selecting a white or blue cushion is $\frac{2}{3}$, which corresponds to Option A.
| Cushion Color | Quantity |
|---|---|
| White | 3 |
| Red | 4 |
| Blue | 5 |
| Total | 12 |
Revision Table: Key Probability Concepts
| Concept | Description | Formula (Basic) |
|---|---|---|
| Probability | The likelihood of an event occurring. | $\text{P(Event)} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$ |
| Mutually Exclusive Events | Events that cannot happen at the same time. | If A and B are mutually exclusive, P(A and B) = 0 |
| 'OR' Rule (Mutually Exclusive) | Probability of either one or the other of two mutually exclusive events occurring. | P(A or B) = P(A) + P(B) |
Additional Information: Probability 'OR' vs 'AND'
In probability, the word 'or' usually implies addition, especially for mutually exclusive events as seen in this cushion problem. The 'or' means we are interested in the total chance of either outcome happening.
The word 'and' usually implies multiplication. For example, if we were drawing two cushions with replacement and wanted the probability of selecting a white cushion first and a blue cushion second, we would multiply the individual probabilities: P(White then Blue) = P(White) × P(Blue) = $\frac{3}{12} \times \frac{5}{12}$. If the draws were without replacement, the second probability would change based on the first draw.
Understanding whether events are mutually exclusive (cannot happen together) or independent (one event doesn't affect the other) is crucial for applying the correct probability rules.
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?
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.
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:
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?
