Probability of Same Color Opposite Side Card Problem

Understanding the Card Scenarios

We have three distinct cards:

• Card 1: Both sides are Black (BB).
• Card 2: Both sides are Red (RR).
• Card 3: One side is Black and one side is Red (BR).

Analyzing the Observation Process

A card is chosen randomly, and one side is observed. We need to find the probability that the side facing down (the opposite side) is the same color as the side we observed.

Let's consider all possible equally likely outcomes based on the side we observe. There are 6 sides in total across the three cards, and any of these could be the one we observe.

Identifying Favorable Outcomes

We list each possible observation and check if the opposite side matches the observed color:

• Scenario 1: Observe a Black side from the BB card. There are 2 Black sides on this card. The opposite side is also Black. (Observed: Black, Opposite: Black) -> Match.
• Scenario 2: Observe a Red side from the RR card. There are 2 Red sides on this card. The opposite side is also Red. (Observed: Red, Opposite: Red) -> Match.
• Scenario 3: Observe a Black side from the BR card. There is 1 Black side on this card. The opposite side is Red. (Observed: Black, Opposite: Red) -> No Match.
• Scenario 4: Observe a Red side from the BR card. There is 1 Red side on this card. The opposite side is Black. (Observed: Red, Opposite: Black) -> No Match.

To calculate the probability, we consider each of the 6 sides as a potential observation.

• The BB card offers 2 possible Black observations. Both result in the opposite side being Black (same color).
• The RR card offers 2 possible Red observations. Both result in the opposite side being Red (same color).
• The BR card offers 1 Black observation (opposite is Red) and 1 Red observation (opposite is Black). Neither results in the same color.

Calculating the Probability

Total number of equally likely observable sides = 2 (from BB) + 2 (from RR) + 2 (from BR) = 6.

Number of favorable outcomes (where the opposite side is the same color as the observed side):

• From the BB card: 2 outcomes.
• From the RR card: 2 outcomes.
• From the BR card: 0 outcomes.

Total favorable outcomes = 2 + 2 = 4.

The probability is calculated as:

$ P(\text{Opposite side is same color}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} $

$ P(\text{Opposite side is same color}) = \frac{4}{6} $

Simplifying the fraction:

$ P(\text{Opposite side is same color}) = \frac{2}{3} $