Question

A coin is loaded so that the probability of a head occurring on a single toss is 2/3. In five tosses of the coin, what is the probability of getting all heads or all tails? The probability of all heads or all tails is (Round to three decimal places as needed.)

Answer 1

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.

Answer 2

To find the probability of getting all heads or all tails in five tosses of a loaded coin, we can use the binomial probability formula. The probability of getting all heads or all tails is 0.263, when rounded to three decimal places.

Step-by-step explanation:

To find the probability of getting all heads or all tails in five tosses of a loaded coin, we can use the binomial probability formula. The probability of getting all heads or all tails in a single toss is 2/3. Let's calculate the probability for getting all heads:

Probability of getting a head in a single toss: 2/3

Probability of getting all heads in five tosses: (2/3) * (2/3) * (2/3) * (2/3) * (2/3)

Calculating the probability: (2/3)^5 = 0.1316

The probability of getting all tails will be the same, since the coin is loaded. Therefore, the probability of getting all heads or all tails in five tosses is 0.1316 + 0.1316 = 0.2632, which is approximately 0.263 when rounded to three decimal places.