Question

Each lottery ticket has a 1 in a million chance of winning, independent from all other tickets (so it is possible to have μltiple winning tickets). 4 million people in California each buy 1 ticket. (a) Find the exact expression for the probability that at least 3 people from California win.

a) (1 - 999,9991,000,000^4,000,000)
b) (11,000,000 × 999,9991,000,000^3,999,997)
c) (1 - left(999,9991,000,000right)^4,000,000)
d) (999,9991,000,000^4,000,000)

Answer 1

The expression for the probability that at least 3 people from California win is (1 - (999,999/1,000,000)^4,000,000), option c.

Step-by-step explanation:

To find the probability that at least 3 people from California win, we can use the complement rule.

Given:

• Each lottery ticket has a 1 in a million chance of winning.
• 4 million people in California each buy 1 ticket.

To find the probability that at least 3 people win, we need to calculate the probability that 0, 1, or 2 people win, and then subtract it from 1 (since the complement rule states that P(A') = 1 - P(A)).

Calculations:

Probability that 0 people win: (999,999/1,000,000)^4,000,000

Probability that 1 person wins: (4,000,000)(1/1,000,000)(999,999/1,000,000)^3,999,999

Probability that 2 people win: (4,000,000 choose 2)(1/1,000,000)^2(999,999/1,000,000)^3,999,998

Now, we can find the probability that at least 3 people win:

P(at least 3 people win) = 1 - P(0 people win) - P(1 person wins) - P(2 people win)

Comparing the given answer choices:

• a) (1 - 999,999/1,000,000^4,000,000) - Incorrect, missing the exponent for 4,000,000.
• b) (11,000,000 × 999,999/1,000,000^3,999,997) - Incorrect, incorrect calculation.
• c) (1 - (999,999/1,000,000)^4,000,000) - Correct, matches the probability expression.
• d) (999,999/1,000,000^4,000,000) - Incorrect, missing 1 - in front.

Therefore, the correct expression for the probability that at least 3 people from California win is (1 - (999,999/1,000,000)^4,000,000), option c.