Question

In 10 independent rollings of a biased die, the probability that an even number will appear 5 times is twice the probability that an even number will appear 4 times. What is the probability that an even number will appear twice when the die is rolled 8 times?

Answer 1

Answer:

700 * 3⁶ / 8⁸

0.0304

Step-by-step explanation:

Probability of even number  p = p

Then probability of odd number (or not even number) = q = 1 -  p

Probability Appearing 5 times

¹⁰C₅ * p⁵ * q¹⁰⁻⁵  

= ¹⁰C₅ * p⁵ * q⁵  

Probability Appearing 5 times

¹⁰C₄ * p⁴ * q¹⁰⁻⁴  

= ¹⁰C₄ * p⁴ * q⁶

¹⁰C₅ * p⁵ * q⁵    = 2   * ¹⁰C₄ * p⁴ * q⁶

=>p * 10!/5!5!  = 2q * 10!/6!4!

=> p * 6 = 2q * 5

=> 3p = 5q

=> 3p = 5(1 - p)

=> 8p = 5

=> p = 5/8

& q = 1 - 5/8 = 3/8

probability that an even number will appear twice when the die is rolled 8 times

= ⁸C₂ * p²q⁸⁻²

= 28  * (5/8)² (3/8)⁶

= 28 * 25 * 3⁶ / 8⁸

= 700 * 3⁶ / 8⁸

= 5,10,300/1,67,77,216

= 0.0304