Final answer:
The probability of getting all heads or all tails in five tosses of a loaded coin, with a 2/3 chance of heads each toss, is approximately 0.136 when rounded to three decimal places.
Step-by-step explanation:
The probability of getting all heads or all tails in five tosses of a loaded coin where heads have a probability of 2/3 and tails have a probability of 1/3 can be calculated separately for each scenario and then added together as they are mutually exclusive events.
For all heads, the probability is (2/3)5 because each toss is independent and the probability of heads must be multiplied for each toss. Similarly, for all tails, the probability is (1/3)5.
Now, we compute both:
Probability of all heads: (2/3)5 = 32/243
Probability of all tails: (1/3)5 = 1/243
The total probability of getting either all heads or all tails is the sum of the two probabilities:
Total Probability = (32/243) + (1/243) = 33/243 ≈ 0.136
When rounded to three decimal places, the probability is 0.136.