> ## Documentation Index > Fetch the complete documentation index at: https://v1-learn.neoartd.my.id/llms.txt > Use this file to discover all available pages before exploring further. # Bernoulli Trials and the Binomial Distribution ## Bernoulli Trial Each performance of an experiment with two possible outcomes is called a **Bernoulli trial**. > Setiap pelaksanaan percobaan dengan dua kemungkinan hasil disebut **percobaan Bernoulli**. In general, a possible outcome of a Bernoulli trial is called a `success` or a `failure`. > Secara umum, hasil yang mungkin dari uji coba Bernoulli disebut `sukses` atau `gagal`. If **p is the probability of a success** and **q is the probability of a failure**, it follows that `p + q = 1`. > Jika **p adalah probabilitas sukses** dan **q adalah probabilitas kegagalan**, maka `p + q = 1`. ### Theorem The probability of exactly k successes in n independent Bernoulli trials, with probability of success p and probability of failure $q = 1 - p$, is Probabilitas tepat k keberhasilan dalam n uji coba Bernoulli independen, dengan probabilitas keberhasilan p dan probabilitas kegagalan $q = 1 - p$, adalah **$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C(n, k) p^k q^{n-k}$** ### Example 1 A coin is biased so that the probability of heads is 2/3. What is the probability that exactly four heads come up when the coin is flipped seven times, assuming that the flips are independent? Koin bias sehingga probabilitas kepala adalah 2/3. Berapa probabilitas bahwa tepat empat kepala muncul ketika koin dibalik tujuh kali, dengan asumsi bahwa koin dilempar secara bebas? **Solution:** * There are $2^7 = 128$ possible outcomes when a coin is flipped seven times. * The number of ways four of the seven flips can be heads is $C(7, 4)$. * Because the seven flips are independent, the probability of each of these outcomes (four heads and three tails) is $(2/3)^4 (1/3)^3$. * Consequently, the probability that exactly four heads appear is
* $C(7, 4) (\dfrac{2}{3})^4 (\dfrac{1}{3})^3 = \dfrac{35 \times 16}{3^7} = \dfrac{560}{2187} \approx 0.256$. ## Binomial Distribution We denote by *$b(k; n, p)$* the probability of k successes in n independent Bernoulli trials with probability of success p and probability of failure `q = 1 - p`. > Kami menyatakan dengan *$b(k; n, p)$* probabilitas keberhasilan k dalam n percobaan Bernoulli independen dengan probabilitas keberhasilan p dan probabilitas kegagalan `q = 1 - p`. Considered as a function of k, we call this function the **binomial distribution** > Dianggap sebagai fungsi k, kita menyebut fungsi ini sebagai **distribusi binomial** $b(k; n,p) = C(n,k) p^k q^{n - k}$ ### Example 2 Suppose that the probability that a 0 bit is generated is 0.9, that the probability that a 1 bit is generated is 0.1, and that bits are generated independently. What is the probability that exactly eight 0 bits are generated when 10 bits are generated? Misalkan probabilitas 0 bit dihasilkan adalah 0,9, probabilitas 1 bit dihasilkan adalah 0,1, dan bit dihasilkan secara independen. Berapa probabilitas bahwa tepat delapan 0 bit dihasilkan ketika 10 bit dihasilkan? **Solution:** * By Theorem 2, the probability that exactly eight 0 bits are generated is
* $b(8; 10, 0.9) = C(10, 8) (0.9)^8 (0.1)^2 = 0.1937102445$. ## Random Variables A random variable is a function from the sample space of an experiment to the set of real numbers. > Variabel acak adalah fungsi dari ruang sampel percobaan ke himpunan bilangan real. A random variable assigns a real number to each possible outcome. > Variabel acak memberikan bilangan real untuk setiap kemungkinan hasil. The distribution of a random variable X on a sample space S is the set of pairs $(r, p(X = r))$ for all $r \in X(S)$, where $p(X = r)$ is the probability that X takes the value r. > Distribusi variabel acak X pada ruang sampel S adalah himpunan pasangan $(r, p (X = r))$ untuk semua $r \in X (S)$, di mana $p(X = r)$ adalah probabilitas bahwa X mengambil nilai r. #### Example 3 * Suppose that a coin is flipped three times. * Let $X(t)$ be the random variable that equals the number of heads that appear when t is the outcome. * Then $X(t)$ takes on the following values: $X(HHH) = 3,$\ $X(HHT) = X(HTH) = X(THH) = 2,$\ $X(TTH) ~= X(THT) = X(HTT) = 1,$\ $X(TTT) ~~= 0.$ #### Example 4 Let $X$ be the sum of the numbers that appear when a pair of dice is rolled. What are the values of this random variable for the 36 possible outcomes $(i,j)$, where $i$ and $j$ are the numbers that appear on the first die and the second die, respectively, when these two dice are rolled? Misalkan $X$ adalah jumlah angka yang muncul saat sepasang dadu dilempar. Berapa nilai variabel acak ini untuk 36 hasil yang mungkin $(i,j)$, di mana $i$ dan $j$ masing-masing adalah angka yang muncul pada dadu pertama dan dadu kedua, ketika kedua dadu ini dilempar? **Solution:** $X((1,1)) = 2,$\ $X((1,2)) = X((2,1)) = 3,$\ $X((1,3)) = X((2,2)) = X((3,1)) = 4,$\ $X((1,4)) = X((2,3)) = X((3,2)) = X((4,1)) = 5,$\ $...$ ### Distribution of Random Variable The **distribution** of a random variable $X$ on a sample space $S$ is the set of pairs $(r, p(X = r))$ for all $r \in X(S)$, where $p(X = r)$ is the probability that $X$ takes the value $r$ The set of pairs in this distribution is determined by the probabilities $p(X = r)$ for $r \in X(S)$ #### Example 5 Suppose that a coin is flipped three times. Let $X(t)$ be the random variable that equals the number of heads that appear when t is the outcome. Determine the distribution of $X(t)$. Misalkan sebuah koin dilempar tiga kali. Misalkan $X(t)$ adalah variabel acak yang sama dengan jumlah kepala yang muncul ketika t adalah hasilnya. Tentukan distribusi $X(t)$. **Solution:** Each of the eight possible outcomes when a fair coin is flipped three times has probability 1/8
$P(X = 3) = 1/8, P(X = 2) = 3/8, P(X = 1) = 3/8$, and $P(X = 0) = 1/8$
Consequently, the distribution of $X(t)$ is the set of pairs $(3,1/8)$, $(2,3/8)$, $(1,3/8)$, and $(0,1/8)$.