BINOM.DIST in Excel

BINOM.DIST in Excel Calculations with the binomial distribution formula can quite tedious and difficult. The reason for this is due to the number and types of terms in the formula.  As with many calculations in probability, Excel can be utilized to expedite the process. Background on the Binomial Distribution The binomial distribution is a discrete probability distribution. In order to use this distribution, we need to make sure that the following conditions are met: There are a total of n independent trials.  Each of these trials can be classified as a success or failure.The probability of success is a constant p. The probability that exactly k of our n trials are successes is given by the formula: C( n, k) pk (1 - p)n – k. In the above formula, the expression C( n, k) denotes the binomial coefficient. This is the number of ways to form a combination of k elements from a total of n. This coefficient involves the use of the factorial, and so C(n, k) n!/[k!(n – k)! ]. COMBIN Function The first function in Excel related to the binomial distribution is COMBIN. This function calculates the binomial coefficient C( n, k), also known as the number of combinations of k elements from a set of n. The two arguments for the function are the number n of trials and k the number of successes. Excel defines the function in terms of the following: COMBIN(number, number chosen) Thus if there are 10 trials and 3 successes, there are a total of C(10, 3) 10!/(7!3!) 120 ways for this to occur. Entering COMBIN(10,3) into a cell in a spreadsheet will return the value 120. BINOM.DIST Function The other function that is important to know about in Excel is BINOM.DIST. There are a total of four arguments for this function in the following order: Number_s is the number of successes. This is what we have been describing as k.Trials are the total number of trials or n.Probability_s is the probability of a success, which we have been denoting as p.Cumulative uses an input either of true or false to calculate a cumulative distribution. If this argument is false or 0, then the function returns the probability that we have exactly k successes. If the argument is true or 1, then the function returns the probability that we have k successes or less. For example, the probability that exactly three coins out of 10 coin flips are heads is given by BINOM.DIST(3, 10, .5, 0). The value returned here is 0.11788. The probability that from flipping 10 coins at most three are heads is given by BINOM.DIST(3, 10, .5, 1). Entering this into a cell will return the value 0.171875. This is where we can see the ease of using the BINOM.DIST function. If we did not use software, we would add together the probabilities that we have no heads, exactly one head, exactly two heads or exactly three heads. This would mean that we would need to calculate four different binomial probabilities and add these together. BINOMDIST Older versions of Excel use a slightly different function for calculations with the binomial distribution. Excel 2007 and earlier use the BINOMDIST function. Newer versions of Excel are backward compatible with this function and so BINOMDIST is an alternate way to calculate with these older versions.

How Chemical Hand Warmers Work

How Chemical Hand Warmers Work If your fingers are cold or your muscles ache, you can use chemical hand warmers to heat them up. There are two types of chemical hand warmer  products, both using exothermic (heat-producing) chemical reactions. Here is how they work. Key Takeaways: Chemical Hand Warmers Chemical hand warmers rely on exothermic chemical reactions to release heat.There are two main types of chemical hand warmers. One type releases heat by air activation. The other type releases heat when a supersaturated solution crystallizes.Air-activated hand warmers are single use products. Chemical solution hand warmers are re-usable. How Air Activated Hand Warmers Work Air-activated hand warmers are long-lasting chemical hand warmers that start working as soon as you unseal the packaging, exposing it to oxygen in​ the  air. The packets of chemicals produce heat from oxidizing iron into iron oxide (Fe2O3) or rust. Each packet contains iron,  cellulose (or sawdust to bulk up the product), water, vermiculite (serves as a water reservoir), activated carbon (distributes heat uniformly),  and salt (acts as a catalyst). This type of hand warmer produces heat anywhere from 1 to 10 hours. Its common to shake the packets to improve circulation, which speeds the reaction and increases the heat. Its possible to get a burn from direct contact between the hand warmer and skin, so the packaging warns users to put the product on the  outside a sock or glove and to keep the packets away from children, who could get burned more easily. Air-activated hand warmers cannot be re-used once they have stopped heating. How Chemical Solution Hand Warmers Work The other type of chemical hand warmer relies on crystallization of a supersaturated solution. The crystallization process releases heat. These hand warmers dont last as long (usually 20 minutes to 2 hours), but they are re-usable. The most common chemical inside this product  is a supersaturated solution of sodium acetate in water. The product is activated by flexing a small metal disk or strip, which acts as a nucleation surface for crystal growth. Usually, the metal is stainless steel.  As the sodium acetate crystallizes, heat is released (up to 130 degrees Fahrenheit). The product can be recharged by heating the pad in boiling  water, which dissolves the crystals back into the small amount of water. Once the package cools, it is ready to use again. Sodium acetate is a food-grade, non-toxic chemical, but other chemicals can be used. Some chemical hand warmers use supersaturated calcium nitrate, which is also safe. Other Types of Hand Warmers In addition to chemical hand warmers, you can get battery-operated hand warmers and also products that work by burning lighter fluid or charcoal inside special cases. All of the products are effective. Which you choose depends on the temperature you want, how long you need the heat to last, and whether you need to be able to re-charge the product. How to Make a Chemical Hand Warmer Its easy to make a DIY hand warmer using iron, salt, and water in a plastic bag. Materials Iron filingsSalt (sodium chloride)Warm (not hot) waterSand, sawdust, vermiculite, or sodium polyacrylate gelZip-top plastic bags Procedure In a small zip-top bag, mix 1-1/2 tablespoons iron filings, 1-1/1 tablespoons salt, 1-1/2 tablespoons sand (or other absorbent material), and 1-1/2 tablespoons warm water.Squeeze the air out of the plastic bag and seal it.Its a good idea to place the bag of chemical inside another bag, remove the excess air, and seal it.Shake or squeeze the content of the bag for about 30 seconds to mix the contents and form a slush. The bag will get hot and will remain hot as long as the chemical reaction proceeds. If the bag gets too hot to hold, set it down. Dont get burned! Another option is to wrap the bag in a sock or towel. This is an air-activated hand warmer. Even though most of the air is squeezed out, enough remains in the bag for the oxidation reaction. If you closely examine the contents of the bag after the reaction is complete, youll see the iron has changed into iron oxide or rust. This type of reaction cannot be reversed unless energy is added, so the hand warmer cannot be re-used. If you want to prepare a homemade hand warmer for later use, keep the salt and water separate from the iron and filler until youre ready for the reaction to occur.