Magic Bag
In a magic bag containing black and white balls, the challenge is to determine the number of white balls present based on the probability of drawing two balls of the same color. The solution involves exploring combinations of black and white balls to arrive at valid solutions. The content delves into calculating probabilities and identifying patterns in consecutive triangular numbers to solve the problem effectively.
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Presentation Transcript
Magic Bag Source: http://nrich.maths.org/7061
Magic Bag A magic bag contains some black and white balls, all of the same size, weight and shape. I put my hands in the bag and simultaneously and randomly pull out two balls. They are both the same colour and the magic bag tells me that the chance of this occurring was exactly 0.5. ? ?? The bag contains a given number of black balls. Can you work out how many white ones there are? What valid combinations of black and white balls are possible? = ? = 435
Magic Bag - Answers 36 45 55 55 66 78 78 91 105 105 120 136 136 153 171 171 190 210 210 231 253 253 276 300 300 325 351 351 378 406 406 435 465 465 496 528 528 561 595 595 630 666
= ? = ? Let p(2 balls identical) = p (b,b or w,w) Since the probabilities are mutually exclusive we can write: ? ?,? + ? ?,? =1 2 ? ? 1 ?+? 1+ ? ? 1 ?+? 1=1 ?+? ?+? 2 2? ? 1 + 2? ? 1 = ? + ? ? + ? 1 2?2 2? + 2?2 2? = ?2+ ?? ? + ?? + ?2 ? ?2 2?? + ?2 ? ? = 0 ? ?2= ? + ?
? ?2= ? + ? A bit of thinking should convince you that the above is satisfied if ? and ? are consecutive triangular numbers.
?? ?? 1 ? ? 1 ? ?? =1 2? ? + 1
Subtracting consecutive triangular numbers ?? ?? 1= ? ?
Adding consecutive triangular numbers ? ??+ ?? 1= ?2 ?
? ?2= ? + ? 2= ??+ ?? 1 ?? ?? 1 But what if we didn t know about triangular numbers? Note the symmetry in the equation if we swap ? and ? the equation remains the same. Let s treat ? as a constant and see what relation ? has to it. We can write a quadratic equation in ?: ?2 2? + 1 ? + ? ? 1 = 0 2? + 12 4? ? 1 2 2? + 1 Another clue about triangular numbers! ? = ? =2? + 1 8? + 1 2 For ? to be a natural number 8? + 1 must be an odd number.
For ? to be a natural number 8? + 1 must be an odd number. Let s call this number 2? + 1. So, 8? + 1 = 2? + 1 8? + 1 = 4?2+ 4? + 1 8? = 4? ? + 1 ? =1 2? ? + 1 So ? must be a triangular number. For ? =1 2? ? + 1 : ? =?2+ ? + 1 2? + 1 2 ? =?2 ? 2? 1 ? or ? =?2+3?+2 =1 =1 2? + 1 ? + 2 2 2 So, when ? = ?? then ? = ?? 1 or ? = ??+1
Note to Teacher There are two answers the number of black balls is a triangular number and the number of white balls are the previous and next triangular numbers The number of black balls are alternate triangular numbers so students should get some answers the same Some students may spot the pattern of numbers from the lower three but they still need to prove it!
Magic Bag A magic bag contains some black and white balls, all of the same size, weight and shape. I put my hands in the bag and simultaneously and randomly pull out two balls. They are both the same colour and the magic bag tells me that the chance of this occurring was exactly 0.5. ? ?? The bag contains a given number of black balls. Can you work out how many white ones there are? What valid combinations of black and white balls are possible? = ? = 45 SIC_10
Magic Bag A magic bag contains some black and white balls, all of the same size, weight and shape. I put my hands in the bag and simultaneously and randomly pull out two balls. They are both the same colour and the magic bag tells me that the chance of this occurring was exactly 0.5. ? ?? The bag contains a given number of black balls. Can you work out how many white ones there are? What valid combinations of black and white balls are possible? = ? = 66 SIC_10
Magic Bag A magic bag contains some black and white balls, all of the same size, weight and shape. I put my hands in the bag and simultaneously and randomly pull out two balls. They are both the same colour and the magic bag tells me that the chance of this occurring was exactly 0.5. ? ?? The bag contains a given number of black balls. Can you work out how many white ones there are? What valid combinations of black and white balls are possible? = ? = 91 SIC_10
Magic Bag A magic bag contains some black and white balls, all of the same size, weight and shape. I put my hands in the bag and simultaneously and randomly pull out two balls. They are both the same colour and the magic bag tells me that the chance of this occurring was exactly 0.5. ? ?? The bag contains a given number of black balls. Can you work out how many white ones there are? What valid combinations of black and white balls are possible? = ? = 120 SIC_10
Magic Bag A magic bag contains some black and white balls, all of the same size, weight and shape. I put my hands in the bag and simultaneously and randomly pull out two balls. They are both the same colour and the magic bag tells me that the chance of this occurring was exactly 0.5. ? ?? The bag contains a given number of black balls. Can you work out how many white ones there are? What valid combinations of black and white balls are possible? = ? = 153 SIC_10
Magic Bag A magic bag contains some black and white balls, all of the same size, weight and shape. I put my hands in the bag and simultaneously and randomly pull out two balls. They are both the same colour and the magic bag tells me that the chance of this occurring was exactly 0.5. ? ?? The bag contains a given number of black balls. Can you work out how many white ones there are? What valid combinations of black and white balls are possible? = ? = 190 SIC_10
Magic Bag A magic bag contains some black and white balls, all of the same size, weight and shape. I put my hands in the bag and simultaneously and randomly pull out two balls. They are both the same colour and the magic bag tells me that the chance of this occurring was exactly 0.5. ? ?? The bag contains a given number of black balls. Can you work out how many white ones there are? What valid combinations of black and white balls are possible? = ? = 231 SIC_10
Magic Bag A magic bag contains some black and white balls, all of the same size, weight and shape. I put my hands in the bag and simultaneously and randomly pull out two balls. They are both the same colour and the magic bag tells me that the chance of this occurring was exactly 0.5. ? ?? The bag contains a given number of black balls. Can you work out how many white ones there are? What valid combinations of black and white balls are possible? = ? = 276 SIC_10
Magic Bag A magic bag contains some black and white balls, all of the same size, weight and shape. I put my hands in the bag and simultaneously and randomly pull out two balls. They are both the same colour and the magic bag tells me that the chance of this occurring was exactly 0.5. ? ?? The bag contains a given number of black balls. Can you work out how many white ones there are? What valid combinations of black and white balls are possible? = ? = 325 SIC_10
Magic Bag A magic bag contains some black and white balls, all of the same size, weight and shape. I put my hands in the bag and simultaneously and randomly pull out two balls. They are both the same colour and the magic bag tells me that the chance of this occurring was exactly 0.5. ? ?? The bag contains a given number of black balls. Can you work out how many white ones there are? What valid combinations of black and white balls are possible? = ? = 378 SIC_10
Magic Bag A magic bag contains some black and white balls, all of the same size, weight and shape. I put my hands in the bag and simultaneously and randomly pull out two balls. They are both the same colour and the magic bag tells me that the chance of this occurring was exactly 0.5. ? ?? The bag contains a given number of black balls. Can you work out how many white ones there are? What valid combinations of black and white balls are possible? = ? = 435 SIC_10
Magic Bag A magic bag contains some black and white balls, all of the same size, weight and shape. I put my hands in the bag and simultaneously and randomly pull out two balls. They are both the same colour and the magic bag tells me that the chance of this occurring was exactly 0.5. ? ?? The bag contains a given number of black balls. Can you work out how many white ones there are? What valid combinations of black and white balls are possible? = ? = 496 SIC_10
Magic Bag A magic bag contains some black and white balls, all of the same size, weight and shape. I put my hands in the bag and simultaneously and randomly pull out two balls. They are both the same colour and the magic bag tells me that the chance of this occurring was exactly 0.5. ? ?? The bag contains a given number of black balls. Can you work out how many white ones there are? What valid combinations of black and white balls are possible? = ? = 561 SIC_10
Magic Bag A magic bag contains some black and white balls, all of the same size, weight and shape. I put my hands in the bag and simultaneously and randomly pull out two balls. They are both the same colour and the magic bag tells me that the chance of this occurring was exactly 0.5. ? ?? The bag contains a given number of black balls. Can you work out how many white ones there are? What valid combinations of black and white balls are possible? = ? = 630 SIC_10
