1518 w A particle undergoes collisions with other particles

15.18 (w) A particle undergoes collisions with other particles. Each collision causes its horizontal velocity to change according to a N(0, 0.1) cm/sec random vari able. After 100 independent collisions what is the probability that the parti Cle\'s velocity will exceed 5 cm/sec if it is initially at rest? Is this result exact or approximate?

Solution

This is like asking the probability of a mean change of 5/100 cm/s = 0.05 cm/s after 100 collisions.

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    0.05      
u = mean =    0      
n = sample size =    100      
s = standard deviation =    0.1      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    5      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   5   ) =    2.86652*10^-7 [ANSWER]

This result is an approximate.

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