Will the next call be a sales call?

Question

A phone line receives sales calls as a Poisson process at rate 2 per hour and support calls independently at rate 8 per hour. What is the probability that the next call to come in is a sales call?

Accepted Answer

Sales and support calls are independent Poisson processes racing to ring first. The sales stream wins in proportion to its rate: P(	ext{sales first}) = \frac{\lambda{	ext{sales}}}{\lambda{	ext{sales}} + \lambda{	ext{support}}} = \frac{2}{2 + 8} = \frac{2}{10} = 0.2. A 20\% chance. Support calls are four times as frequent, so they win four out of five times, leaving sales one in five. Takeaway. Rate share sets the odds of who arrives first; a 2\!:\!8 rate split makes the next call a sales call just 20\% of the time.

Will the next call be a sales call?

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