| Abdullah Al Mahmud | www.statmania.info |

- Success/failure
- Detection of a disease is a success!

The probability of choosing statistics is 0.16. If 20 students are randomly selected, what is the probability that

- at least 5 would choose the subject?
- exactly 2 would choose the subject?
- none will choose?

\(P(x) = ^nC_x \space p^x(1-p)^{n-x}\)

\(1-p=q\)

- \(p \to\) probability of success
- \(q \to\) probability of failure
- \(n \to\) number of Bernoulli trials
- \(x \to\) no. of success

- Introduced by James Bernoulli
- Independent trials with dichotomous outcomes (success/failure)
- Coin toss, disease detection

\(p^xq^{1-x}\)

If n = 1 in binomial distribution

- Trials are independent
- p, q remain constant for each trial
- Outcomes are dichotomous
- X counts number of success

- Mean: \(np\)
- Variance: \(npq\)
- Skewness, \(\beta_1 = \frac{(q-p)^2}{npq}\)
- Kurtosis, \(\beta_2 = 3 + \frac{1-6pq}{npq}\)

\(P(SF SF \cdots) = P(S) \times P(F) \cdots\)

- \((p \times p \cdots \times p)(\times q \times q \cdots \times q)\)
- \(p^xq^{n-x}\)
- \(^nC_x \space p^xq^{n-x}\) (any x combination from n)
- 2 items from a, b, c \(\to ab, ac, bc \to \space ^3C_2\)

\(\displaystyle E(X) = \sum_{x=0}^n x\cdot p(x)\) (think: why zero?)

- \(0 \cdot p(0) + \cdots\)
- \(np\{q^{n-1}+n(n-1)pq^{n-2}+\cdots+p^{n-1}\}\)
- \(np(q+p)^{n-1} [binomial \ theorem]\)
- \(np\) [p+q=1]

- Quality control
- Test of fit

If \(n \to \infty\) and \(np\ge 5, nq \ge 5\)

The mean and sd of a binomial distribution are 40 and 6. Find

- \(n,p,q\)
- \(P(x)\)
- \(P(x=0), P(x\ge2), P(x\lt3)\)

- \(np = 40\)
- \(\sqrt{npq} = 6\)
- \(np(1-p)=36 \to 40(1-p)=36 \to 1-p=.9\)

An unbiased coin is tossed 10 times. Find the probability of

- 3 heads
- at least 2 heads
- at most 1 head

The probability that it rains on a particular day is 0.2. If 10 days are observed, what is the probability that it would rain on

- all 10 days
- no days
- at least 2 days
- on exactly four days

In El Clasico, the probability that Barcelona wins is 0.39 and that Real Madrid wins is 0.42. In the next 5 matches, what is the probability that

- Barcelona wins all the matches?
- Real wins all the matches
- Barcelona wins at least 2 matches
- 2 matches are drawn
- more than 3 matches are drawn

- \(p_1=\)Barca wins, \(p_2 =\) Real wins, \(p_3 =\)Match drawn