Understanding the Probability Problem
The problem asks for the probability of selecting two balls of the same colour when drawing two balls at random from a bag. The bag contains a specific number of black and white balls.
Key Information:
- Number of black balls = 5
- Number of white balls = 4
- Total number of balls = 5 + 4 = 9
- Number of balls selected = 2
- Selection is done at random.
Calculating Total Possible Outcomes
We need to find the total number of ways to select any 2 balls from the 9 balls available. Since the order of selection doesn't matter, we use combinations. The formula for combinations is \(C(n, k) = \frac{n!}{k!(n-k)!}\), where \(n\) is the total number of items, and \(k\) is the number of items to choose.
Total ways to select 2 balls from 9 is:
\(C(9, 2) = \frac{9!}{2!(9-2)!} = \frac{9!}{2!7!} = \frac{9 \times 8}{2 \times 1} = 36\)So, there are 36 possible pairs of balls that can be selected.
Calculating Favourable Outcomes
The condition is that both selected balls must be of the same colour. This can happen in two mutually exclusive ways:
- Both balls selected are black.
- Both balls selected are white.
Scenario 1: Both Balls are Black
Number of ways to select 2 black balls from the 5 black balls available:
\(C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{5 \times 4}{2 \times 1} = 10\)Scenario 2: Both Balls are White
Number of ways to select 2 white balls from the 4 white balls available:
\(C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{4!}{2!2!} = \frac{4 \times 3}{2 \times 1} = 6\)Determining the Probability
The total number of favourable outcomes is the sum of the outcomes from the two scenarios (both black OR both white):
Total Favourable Outcomes = (Ways to select 2 black balls) + (Ways to select 2 white balls)
\(\text{Total Favourable Outcomes} = 10 + 6 = 16\)The probability of an event is calculated as:
\(P(\text{Event}) = \frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}}\)Therefore, the probability of selecting two balls of the same colour is:
\(P(\text{Same Colour}) = \frac{16}{36}\)Simplifying the fraction by dividing the numerator and denominator by their greatest common divisor, which is 4:
\(P(\text{Same Colour}) = \frac{16 \div 4}{36 \div 4} = \frac{4}{9}\)Similar Questions
- What is the value of \(n\) ?
- What is the mean of the distribution ?
- What is the value of \(p + q\) ?
- What is \(\sum_{i=1}^n x_i f_i\) equal to ?
- What is the mean deviation of the largest five observations ?
- What is the variance of the largest five observations ?
What is the total number of students whose height is less than or equal to 165 cm?
What is the median height of the class?
The height which occurs most frequently in the class is
The most appropriate graphical representation of the given frequency distribution is
Important Questions from Statistics
Match the following:
(a) Marginalist Revolution (i) Samuelson (b) Multiplier-Accelerator model (ii) J. R. Hicks (c) IS-LM curves (iii) Jevous (d) Real Business Cycle (iv) Robert J. Borro Choose the correct option from those given below:
As per the SRS Bulletin of September 2017, the estimated death rate for Kerala is 7.6, while for Bihar it is 6. From these data which is the correct inference to draw?
Arrange the following States in descending order according to Maternal Mortality Ratio (MMR) as per the Special Bulletin of SRS, May, 2018:
(i) Assam
(ii) Bihar
(iii) Madhya Pradesh
(iv) Uttar Pradesh
Choose the correct answer from the code given below :
Which of the following statements is true for the Indian economy according to the World Bank figures for 2017?
Harrod's Growth model is given as under:
\(\begin{array}{ll} \mathrm{S}_{\mathrm{t}}=\alpha \mathrm{Y}_{\mathrm{t}} & 0<\alpha<1 \\ \mathrm{I}_{\mathrm{t}}=\beta\left[\mathrm{Y}_{\mathrm{t}}-\mathrm{Y}_{\mathrm{t}-1}\right] & \beta>0 \\ \mathrm{~S}_{\mathrm{t}}=\mathrm{I}_{\mathrm{t}} & \end{array}\)
where S t = Savings, Y t = Income, l t = Investment, t = time
In this model for economic growth, the condition for economic growth is
