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Math  /  Data & Statistics

QuestionSuppose that you roll a fair, 6-sided die 10 times. What's the probability that you get at least one 6 ? □\square (Round to 4 decimal places. Leave your answer in decimal form.)

Studdy Solution

STEP 1

1. A fair 6-sided die means each side has an equal probability of being rolled.
2. The probability of rolling any specific number (including 6) on a 6-sided die is 16 \frac{1}{6} .
3. The probability of not rolling a 6 on a single roll is 56 \frac{5}{6} .
4. The problem asks for the probability of getting at least one 6 in 10 rolls.

STEP 2

1. Calculate the probability of not rolling a 6 in 10 rolls.
2. Use the complement rule to find the probability of rolling at least one 6.

STEP 3

Calculate the probability of not rolling a 6 on a single roll:
P(not 6 on one roll)=56 P(\text{not 6 on one roll}) = \frac{5}{6}

STEP 4

Calculate the probability of not rolling a 6 in 10 rolls:
P(not 6 in 10 rolls)=(56)10 P(\text{not 6 in 10 rolls}) = \left(\frac{5}{6}\right)^{10}

STEP 5

Use the complement rule to find the probability of rolling at least one 6:
P(at least one 6)=1−P(not 6 in 10 rolls) P(\text{at least one 6}) = 1 - P(\text{not 6 in 10 rolls})

STEP 6

Calculate (56)10 \left(\frac{5}{6}\right)^{10} :
(56)10≈0.1615 \left(\frac{5}{6}\right)^{10} \approx 0.1615

STEP 7

Calculate the probability of rolling at least one 6:
P(at least one 6)=1−0.1615=0.8385 P(\text{at least one 6}) = 1 - 0.1615 = 0.8385
The probability of rolling at least one 6 in 10 rolls is:
0.8385 \boxed{0.8385}

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