Understanding probability and how to handle equations involving variables like ( n ) can be challenging for students, but with the right approach, it’s definitely achievable. In this post, we’ll break down a probability problem step by step, shedding light on how to navigate through the intricacies without getting overwhelmed.
Introduction
So, the question at hand involves probability with variables, commonly seen in GCSE Grade 8_9. The setup is about counters in a bag, and we’ll use algebra to solve it.
There’s ( n ) counters in the bag.
Out of these, 4 are red.
We need to determine the number of blue counters and solve for ( n ).
Problem Breakdown
Setting Up the Problem
Let’s dive into the details:
“So the questions, again, another probability question. So here we’ve got there are ( n ) counters in the bag. So we already know the total number of counters for this, which is perfect for us because that’s going to help us to solve this question because that denominator is going to be ( n ). Right? So we’ve got ( n ) counters.”
From the question, we know:
There are ( n ) counters in total in the bag.
4 out of these ( n ) counters are red.
Straightaway:
Number of blue counters ( = n – 4 )
Setting Up the Probability
“And so ( n ) takeaway 4 is our numerator for the first branch of blue. And then we’ve got another branch because it says the probability of Ross that takes out two counters that are blue is a third. So really, I’m just going to focus my efforts on blue and blue. And so for me to work out the second branch of blue, I know there’ll be one less counter.”
Key Points:
Probability of first blue counter = ( \frac{n – 4}{n} )
Once a blue counter is taken, probability of the second blue counter = ( \frac{n – 5}{n – 1} )
Constructing the Probability Equation
“Now, as we know, to work out probability with tree diagrams for blue and blue, we’re going to multiply the probabilities.”
The probability of drawing two blue counters:[ \frac{(n – 4)}{n} \times \frac{(n – 5)}{n – 1} = \frac{1}{3} ]
Simplifying the Equation
We simplify the expression step by step:
“And so this is just a fraction multiplied by another fraction. What do we about multiplying fractions? We multiply the numerators and denominators. Right? So that’s exactly what I’m going to do.”
[ \frac{(n – 4)(n – 5)}{n(n – 1)} = \frac{1}{3} ]
Expanding and simplifying:[ (n – 4)(n – 5) = n^2 – 9n + 20 ][ n(n – 1) = n^2 – n ]
So we have:[ \frac{n^2 – 9n + 20}{n^2 – n} = \frac{1}{3} ]
Eliminating the Denominator
To eliminate the denominator and solve for ( n ):
“So I’m going to get rid of the three first, right? So I’m going to multiply three on both sides. That means a third times three is a whole, isn’t it?”
[ 3(n^2 – 9n + 20) = n^2 – n ]
Rearrange and Combine Like Terms
Reorganise to look like a quadratic equation:
Expand and simplify:[ 3n^2 – 27n + 60 = n^2 – n ]
Combine like terms:[ 3n^2 – 27n + 60 – n^2 + n = 0 ][ 2n^2 – 26n + 60 = 0 ]
Final Simplification
“Two times the bracket. So in order for me to eliminate the two, … divide by two on both sides.”
[ n^2 – 13n + 30 = 0 ]
Solving the Quadratic Equation
Next, we factorise the quadratic equation:
[ n^2 – 13n + 30 = 0 ]
Find factors that multiply to ( 30 ) and add to ( -13 ):[ (n – 3)(n – 10) = 0 ]
Therefore:[ n = 3 \quad \text{or} \quad n = 10 ]
Logical Consistency
As ( n ) cannot be 3 (because it must be more than 4 since there are 4 red counters), we conclude:
[ n = 10 ]
Conclusion
Solving complex probability questions is all about taking a systematic approach. We’ve shown that:
[ n^2 – 13n + 30 = 0 ][ n = 10 ]
Keep a clear head and follow the logical steps. When facing algebra with fractions and denominators, remember not to panic and tackle it step by step. You’ve got this!
“It’s a bit of a challenging question because there’s a lot of working out, as you can see. But it’s definitely doable, it’s definitely possible, … It’s just taking a systematic approach and not panicking.”
Feel free to try out different combinations and always verify that the logical consistency holds true. Happy studying!
Next:Try your hand at solving Question 7 by yourself!
Did these steps help you? Feel free to leave a comment or question below!