Independent Events: Marbles

Drawing with replacement = independent

Calculator Dice Coins Marbles Cards Learn

Drawing with ReplacementWith Replace Multiple DrawsMulti Draw Same Color TwiceSame Color

Same Color Twice (with Replacement)

What does "same color twice" mean?

The first and second marbles must be the same color. This could happen in two ways: both red OR both blue.

Strategy: Break into cases

"Same color" = "both red" OR "both blue". Since these are mutually exclusive (can't be both red AND both blue at the same time), we:

Calculate P(both red) using multiplication rule
Calculate P(both blue) using multiplication rule
Add them together (union of mutually exclusive events)

Setup:

Bag: 6 red + 4 blue = 10 marbles. Drawing twice with replacement.

red: 6

blue: 4

Total: 10 marbles

Two Ways to Get Same Color

Both Red

Both Blue

Combine Using Union

These events are mutually exclusive (cannot both happen), so we add:

Result:

Comparison: With vs Without Replacement

With replacement, P(same color) = 52%. Without replacement, it would be different because the second draw probability changes based on the first draw.

With Replacement:

P(R,R) = 6/10 × 6/10 = 36/100

Without Replacement:

P(R,R) = 6/10 × 5/9 = 30/90