The Experiment Is To Toss Three Coins For Each Coin The Poss

The Experiment Is To Toss Three Coins For Each Coin The Possible O

The experiment involves various probability scenarios, including tossing coins, rolling dice, selecting balls from a box, and calculating conditional probabilities. The specific questions are as follows:

  1. Determine the probability that exactly one coin will land on tails when tossing three coins, with outcomes {H, T} for each coin.
  2. Calculate the probability that a single roll of a fair six-sided die results in a number 2 or higher.
  3. Find the probability that, when drawing three balls without replacement from a box containing 6 red, 4 yellow, and 10 white balls, exactly zero white balls are drawn.
  4. Determine the probability of drawing three balls without replacement from a box with 20 balls (4 red, 5 yellow, 11 white), such that exactly one ball of each color is selected.
  5. Given probabilities for two events A and B, with P(A) = 0.95, P(B) = 0.51, and P(A ∩ B) = 0.34, compute the conditional probability that A occurs given that B has occurred.

Paper For Above instruction

Understanding probability is fundamental in statistics and everyday decision-making, providing insights into the likelihood of various events. This paper explores different probability scenarios involving coin tossing, die rolling, and drawing balls from a container, along with the concept of conditional probability. These examples illuminate core principles such as combinatorial calculations, independence, and dependence of events, which are essential in probabilistic reasoning and statistical analysis.

Probability of Exactly One Tail in Three Coin Tosses

When tossing three coins, each with outcomes {H, T}, the total number of equally likely outcomes is 2^3 = 8. These outcomes are: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. To find the probability that exactly one coin shows tails, identify the outcomes with exactly one T:

  • HHT
  • HTH
  • THH

There are 3 favorable outcomes out of 8 possible, so the probability is:

P = 3 / 8 = 0.375

Thus, the probability that exactly one coin will be T is 0.375, rounded to three decimal places as 0.375.

Probability of Rolling a 2 or Higher on a Fair Die

A fair six-sided die has outcomes {1, 2, 3, 4, 5, 6}, each with equal probability. To find the probability that the outcome is 2 or higher, enumerate favorable outcomes: 2, 3, 4, 5, 6.

Total favorable outcomes: 5

Total possible outcomes: 6

Therefore, the probability is:

P = 5 / 6 ≈ 0.8333

Rounded to three decimal places, the probability is 0.833.

Probability of Drawing Zero White Balls from a Container of Mixed Balls

In a box with 6 red, 4 yellow, and 10 white balls, the total number of balls is 20. Drawing three balls without replacement, the probability that none of these are white (i.e., all are either red or yellow) can be found using the hypergeometric distribution.

Number of non-white balls: 6 + 4 = 10

Number of white balls: 10

Number of ways to choose 3 balls, all from non-white balls:

C(10, 3)

Number of ways to choose any 3 balls from all 20:

C(20, 3)

Probability:

P = C(10, 3) / C(20, 3) = (120) / (1140) ≈ 0.1053

Rounded to three decimal places, this gives 0.105.

Probability of Drawing One Ball of Each Color in Three Draws

The box contains 20 balls: 4 red, 5 yellow, and 11 white. Drawing three balls without replacement, the probability that exactly one of each color is obtained involves calculating the favorable arrangements over total arrangements.

Number of ways to select one red, one yellow, and one white:

C(4, 1) C(5, 1) C(11, 1) = 4 5 11 = 220

Total number of ways to select 3 balls from 20:

C(20, 3) = 1140

Therefore, the probability is:

P = 220 / 1140 ≈ 0.1930

Rounded to three decimal places, this is 0.193.

Conditional Probability of Event A Given Event B

Given P(A) = 0.95, P(B) = 0.51, and P(A ∩ B) = 0.34, the conditional probability P(A | B) is calculated as:

P(A | B) = P(A ∩ B) / P(B) = 0.34 / 0.51 ≈ 0.6667

Rounded to three decimal places, the conditional probability is 0.667.

Conclusion

The above calculations demonstrate fundamental probability principles through practical examples involving random experiments. Whether tossing coins, rolling dice, or drawing balls from a container, understanding how to enumerate outcomes, calculate ratios of favorable cases to total cases, and interpret conditional probabilities enhances analytical skills essential in statistics and decision sciences.

References

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