Final answer:
The probability of getting exactly 4 heads in 10 tosses of a fair coin is approximately 20.51%. The probability of getting '6' exactly 4 times in 10 rolls of a fair die is approximately 5.43%. This is calculated using the binomial probability formula.
Step-by-step explanation:
The subject of this question is mathematics, specifically the topic of probability, where the student is asked to calculate the probability of getting a certain number of heads when tossing coins, and getting a specific outcome when rolling dice a number of times.
These problems involve the use of the binomial probability formula and combinatorial principles.
Probability of Getting Exactly 4 Heads in 10 Tosses of a Fair Coin
For a fair coin tossed 10 times, the probability of obtaining exactly 4 heads is given by the binomial probability formula:
P(X=k) = C(n, k) × (p^k) × ((1-p)^(n-k))
Where:
C(n, k) is the number of combinations of n items taken k at a time.
p is the probability of getting a head in one toss.
k is the number of heads we want to get (which is 4 in this case).
n is the total number of tosses (which is 10).
For this fair coin p = 0.5, so the probability is:
P(4 heads in 10 tosses) = C(10, 4) × (0.5^4) × (0.5^(10-4)) = 210 × 0.0625 × 0.0625 ≈ 0.2051 or 20.51%
Probability of Getting '6' Exactly 4 Times in 10 Rolls of a Fair Die
Similarly, for a fair six-sided die rolled 10 times, the probability of rolling '6' exactly 4 times can be calculated by using the same formula, with p = 1/6 (since there is one '6' in a die of six faces), thus:
P(6 rolled 4 times in 10 rolls) = C(10, 4) × (1/6)^4 × (5/6)^(10-4) ≈ 0.0543 or 5.43%