A bag contains even and odd numbered balls in the ratio of 3:7, respectively. For each of the following, what is the probability of drawing an even-numbered ball?

a) The total number of balls is 240 and 30 of the odd-numbered balls are renumbered by multiplying the numbers by 4.

b) Half of the odd-numbered balls are renumbered by multiplying the numbers by 6 and a third of the even-numbered balls are renumbered by multiplying the numbers by 5.

c) First, half of the even-numbered balls are renumbered by adding 3 to the numbers, followed by half of the odd-numbered balls being renumbered by adding 5 to the numbers.

a) 0.425

b) 0.65

c) 0.575

Step-by-step explanation:

The ratio of even balls to odd balls is 3:7. That means if there are 3 even balls, then there are 7 odd balls. That makes a total of 10 balls, of which 30% are even and 70% are odd.

a) There are 240 balls, so originally, 72 are even and 168 are odd. 30 of the odd balls are renumbered by multiply by 4. An odd number times an even number is an even number, so 30 of the odd balls become even balls. So there are now 102 even balls and 138 odd balls. Therefore:

P (even) = 102 / 240

P (even) = 0.425

b) Originally, 70% of the balls are odd. Half of those are renumbered by multiplying by 6, so they become even numbers.

P (odd) = 0.5 (70%) = 35%

P (even) = 30% + 35% = 65%

A third of the even balls are renumbered by multiplying by 5, so they remain even numbers.

Therefore, P (even) = 0.65.

c) Originally, 30% of the balls are even. Half of those are renumbered by adding 3. An even number plus an odd number is an odd number, so:

P (even) = 0.5 (30%) = 15%

P (odd) = 70% + 15% = 85%

Next, half the odd balls are renumbered by adding 5. An odd number plus an odd number is an even number, so:

P (odd) = 0.5 (85%) = 42.5%

P (even) = 15% + 42.5% = 57.5%

Therefore, P (even) = 0.575.