Probability of 4 starters in a 7-round draft when p is not constant each round

There are 7 Rounds in a Little League Draft. The probability of drafting a starter in the first three rounds is .55. In the 4th round, the probability of drafting a starter drops to .35. In the 5th, 6th, and 7th rounds of the draft, the probability of drafting a starter drops to .25.

What is the probability that a team will draft at least 4 starters in this draft?

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Hint as I assume this is homework

You can calculate probabilities in each round using the results from the previous round: here are the first few parts

• First round: $Pr(1\text{ starter}) = 0.55\times 1 = 0.55$, $Pr(0\text{ starters}) = 0.45\times 1 = 0.45$
• Second round: $Pr(2\text{ starters}) = 0.55\times 0.55 = 0.3025$, $Pr(1\text{ starter}) = 0.55\times 0.45+0.45\times 0.55 = 0.495$, $Pr(0\text{ starters}) = 0.45\times 0.45 = 0.2025$
• Third round: $Pr(3\text{ starters}) = 0.55\times 0.3025 = 0.166375$, $Pr(2\text{ starters}) = 0.55\times 0.495+0.54\times 0.3025 = 0.408375$, $Pr(1\text{ starter}) = 0.55\times 0.2025+0.45\times 0.495 = 0.334125$, $Pr(0\text{ starters}) = 0.45\times 0.2025 = 0.091125$

Just keep going. You chould get a final answer slightly more than a quarter

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Henry, thank you a ton. I just solved it. It was a lot of fun actually. –  jack Apr 26 '11 at 7:28