If we sample from a small finite population without replacement, the binomial distribution should not be used because the events are not independent. If sampling is done witho replacement and the outcomes belong to one of two types, we can use the hypergeometric distribution. If a population has \( A \) objects of one type, while the remaining \( B \) objects the other type, and if \( n \) objects are sampled without replacement, then the probability of getting \( x \) objects of type \( A \) and \( n-x \) objects of type \( B \) under the hypergeometric distribu given by the following formula. In a lottery game, a bettor selects six numbers from 1 to 51 (without repetition), and a winning six-number combination is later randomly selected the probabilities of getting exactly two winning numbers with one ticket. (Hint: Use \( A=6, B=45, n=6 \), and \( x=2 \) ) \( P(x)=\frac{A!}{(A-x)!x!} \cdot \frac{B!}{(B-n+x)!(n-x)!} \div \frac{(A+B)!}{(A+B-n)!n!} \) \( P(2)=\square \) (Round to four decimal places as needed.)

A three-digit number is to be formed from the digits 1,6 and 8 . What is the probability that the number will:

9.1 Given: \( P(A)=0,45 ; P(B)=y \) and \( P(A \) or \( B)=0,74 \). Determine the value(s) of \( y \) : 9.1.1 If \( A \) and \( B \) are mutually exclusive events. 9.1.2 If \( A \) and \( B \) are independent events.

Topic - Probability A restaurant offers a 3 course meal. The entrée may be garlic bread \( (G) \) or chicken soup (S), the a) Draw a tree diagram and list (sample space) all the possible combinations for a 3 course meal.

Someone hands you a box of 13 chocolate-covered candies, telling you that 6 are vanilla creams and the rest are peanut butter. You pick candies at random and discover the first 3 you eat are all vanilla. Consider a result to be surprising if there is less than a \( 5 \% \) chance of observing it given your expectations. Complete parts a through c below. a) If there really were 6 vanilla and 7 peanut butter candies in the box, what is the probability that you would have picked 3 vanillas in a row? (Round to three decimal places as needed.)

6) รหัสหนังสือของห้องสมุดแห่งหนึ่งประกอบด้วย ตัวอักษรภาษาอังกฤษ 2 ตัว เลขโดด 2 ตัว ที่ไม่เป็นศูนย์พร้อมกัน ตัวอักษร ภาษาอังกฤษ 1 ตัว และเลขโดด 1 ตัวที่ไม่เป็นศูนย์ เช่น \( A B 02 M 3 \) จงหาจำนวนรหัสที่เป็นไปได้ทั้งหมด 7) ถ้าทอดลูกเตำหนึ่งลูกสองครั้ง จงหาจำนวนวิธีที่แต้มที่ได้จากการทอดลูกเต๋าทั้งสองครั้งไม่เท่ากัน